# Computing Limits through Riemann Sums.

In this paper, we will illustrate how to use the idea of Riemann
Sums to evaluate certain difficult limits in calculus. The method we are
about to discuss, even though not entirely new, is missing from most
modern calculus texts (see [5]). It is worthwhile to recall the idea of
Riemann Sums first. The Riemann Sums in general can be defined for a
larger class of functions, but for simplicity we will just consider a
montonically increasing nonnegative function f (x) defined on a closed
interval [0, a], where a is a positive real number. We know that the
area bounded by the graphs of y = f (x), x = 0, x = a, and y = 0 is
given by the following definite integral:

Area = [[integral].sup.a.sub.0] f (x) dx (1)

Let us now divide the interval [0, a] into n subintervals of equal length a/n, where n is any positive integer. One can construct two types of rectangles, using these subintervals as bases. The first kind has heights given by f ((i -- l)a/n), where i = 1, 2, ..., n. As given in Figure 1, the sum of the areas of these rectangles is called a Riemann Lower Sum of the function f (x) over the interval [0, a]. Similarly, a second kind of rectangles have heights given by f (ia/n), where i = 1, 2, ..., n. As given in Figure 2, the sum of the areas of these rectangles is called a Riemann Upper Sum.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

One central idea of integral calculus is that both the Riemann Lower Sum, and the Riemann Upper Sum are approximations for the area under the graph of y = f( x) (see [3] and [4]):

Area = [[integral].sup.a.sub.0] f(x)dx [approximately equal to] [n.summation over (i=1)]a/n f (ai/n) Riemann Upper Sum

Area = [[integral].sup.a.sub.0] f(x)dx [approximately equal to] [n.summation over (i=1)]a/n f (a(i -1)/n) Riemann Lower Sum

For larger and larger n values, the above Riemann Sums will give better approximations for the area. More precisely, the limit of any of the Riemann Sums as n [right arrow] [infinity] to is equal to the true area under the graph.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

The above Figures 1 and 2 only display nonnegative monotonically increasing functions. The same ideas can equally be used for nonnegative monotonically decreasing functions as well.

We will now show how to use the idea of Riemann Sums to calculate some interesting limits.

Example 1 Evaluate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where b is a positive real number.

The idea is to rewrite the above limit as the limit of some Riemann Sum. First, observe the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, our problem is equivalent to finding

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By comparing with equation (3), one can see that the quantity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

represents the limit of the Riemann Lower Sum of the function f (x) = 1/(1+x) over the interval [0,b].

Thus it is equal to the definite integral [[integral].sup.b.sub.0] 1 + 1 + x dx, which can be calculated by elementary calculus (see [3] and [4]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, the required limit is equal to ln(1 + b) /b.

Example 2 Evaluate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

First note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, equation (2) implies that the required limit is equal to the definite integral of the function f (x) = 1/(1 + [x.sup.3]) over the interval [0,1], calculated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The above integral can be either computed by hand, or by using a suitable computer algebra system (CAS). For example, the following command in Maple can be used to compute the required integral (see [1], [6] and [7] ):

> int (1/(1 + x ^ 3), x = 0 .. 1)

Example 3 The reader is now encouraged to prove the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example 4 Calculate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where, m [not equal to] - 1 is a real number.

Using methods similar to above, we can perform the calculation below:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example 5 The reader is also encouraged to calculate the following limit:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example 6 Evaluate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where f (x) is any function such that the integral [[integral].sup.1.sub.0] ln f (x) dx is convergent. Let

u = [nth square root of f (1/n) f(2/n)f(3/n) ... f(n/n).

Taking ln of both sides, we obtain

ln u = 1/n [[n.summation over (i=1) ln f (i/n)].

Therefore, by methods described before, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies that the required limit is equal to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We will record the result as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example 7 Using the above Example 6, the reader can show the following identities:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example 8 Evaluate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We will compute this limit using equation (4) with f( x) = x:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The limit in this example can also be calculated using the following command in Maple:

>limit(factorial(n)^(1/n)/n, n = infinity)

Though Maple uses sophisticated algorithms for finding limits, we cannot expect Maple to calculate any type of limit. In an attempt to calculate the limit

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in Example 4, the following Maple command does not produce an answer:

>limit((sum(k^m, k = 1 .. n))/n^(m+1), n = infinity)

In this paper, we have shown a method of calculating certain limits via suitable Riemann Sums. In some cases we were able to perform the calculation by hand, but in other cases, the calculation was facilitated by a CAS. However, we hope that the paper shed some light into the limitations of a CAS as illustrated by Example 4.

Acknowledgements

The author gratefully appreciates the editors and referees for their valuable comments.

References

[1] Norman Chonacky and David Winch. (2005). Reviews of Maple, Mathematica, and Matlab: Coming Soon to a Publication Near You. Computing in Science and Engineering, 7(2), 9-10.

[2] Yang, Hansheng and Heng Yang. (2001). The Arithmetic-Geometric Mean Inequality and the Constant e. Mathematics Magazine. 74(4): 321-323.

[3] George B. Thomas and Ross L. Finney. Calculus and Analytic Geometry (9th Edition) (Hardcover). Addison Wesley Longman (Higher Education Division, Pearson Education) . ISBN-10: 0201509008.

[4] Howard Anton and Albert Herr. Calculus With Analytic Geometry (Hardcover). John Wiley & Sons; 5 Sub edition ISBN-10: 0471594954.

[5] Sudhir K. Goel; Dennis M. Rodriguez.(1987). A Note on Evaluating Limits Using Riemann Sums. Mathematics Magazine, Vol. 60, No. 4, 225-228.

[6] Richard E. Klima; Neil Sigmon and Ernest Stitzinger. (2007). Applications of abstract algebra with Maple and MATLAB. Chapman andHall/CRC. ISBN: 1-58488-610-2.

[7] Henrik Aratyn and Constantin Rasinariu. (2006). A short course in mathematical methods with Maple. World Scientific Publishing Co. Pte. Ltd. ISBN:981-256-595-7.

Sanjay Kumar Khattri

sanjay.khattri@hsh.no

Department of Engineering

Stord Haugesund University College

5528

Norway

Area = [[integral].sup.a.sub.0] f (x) dx (1)

Let us now divide the interval [0, a] into n subintervals of equal length a/n, where n is any positive integer. One can construct two types of rectangles, using these subintervals as bases. The first kind has heights given by f ((i -- l)a/n), where i = 1, 2, ..., n. As given in Figure 1, the sum of the areas of these rectangles is called a Riemann Lower Sum of the function f (x) over the interval [0, a]. Similarly, a second kind of rectangles have heights given by f (ia/n), where i = 1, 2, ..., n. As given in Figure 2, the sum of the areas of these rectangles is called a Riemann Upper Sum.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

One central idea of integral calculus is that both the Riemann Lower Sum, and the Riemann Upper Sum are approximations for the area under the graph of y = f( x) (see [3] and [4]):

Area = [[integral].sup.a.sub.0] f(x)dx [approximately equal to] [n.summation over (i=1)]a/n f (ai/n) Riemann Upper Sum

Area = [[integral].sup.a.sub.0] f(x)dx [approximately equal to] [n.summation over (i=1)]a/n f (a(i -1)/n) Riemann Lower Sum

For larger and larger n values, the above Riemann Sums will give better approximations for the area. More precisely, the limit of any of the Riemann Sums as n [right arrow] [infinity] to is equal to the true area under the graph.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

The above Figures 1 and 2 only display nonnegative monotonically increasing functions. The same ideas can equally be used for nonnegative monotonically decreasing functions as well.

We will now show how to use the idea of Riemann Sums to calculate some interesting limits.

Example 1 Evaluate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where b is a positive real number.

The idea is to rewrite the above limit as the limit of some Riemann Sum. First, observe the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, our problem is equivalent to finding

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By comparing with equation (3), one can see that the quantity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

represents the limit of the Riemann Lower Sum of the function f (x) = 1/(1+x) over the interval [0,b].

Thus it is equal to the definite integral [[integral].sup.b.sub.0] 1 + 1 + x dx, which can be calculated by elementary calculus (see [3] and [4]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, the required limit is equal to ln(1 + b) /b.

Example 2 Evaluate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

First note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, equation (2) implies that the required limit is equal to the definite integral of the function f (x) = 1/(1 + [x.sup.3]) over the interval [0,1], calculated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The above integral can be either computed by hand, or by using a suitable computer algebra system (CAS). For example, the following command in Maple can be used to compute the required integral (see [1], [6] and [7] ):

> int (1/(1 + x ^ 3), x = 0 .. 1)

Example 3 The reader is now encouraged to prove the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example 4 Calculate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where, m [not equal to] - 1 is a real number.

Using methods similar to above, we can perform the calculation below:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example 5 The reader is also encouraged to calculate the following limit:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example 6 Evaluate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where f (x) is any function such that the integral [[integral].sup.1.sub.0] ln f (x) dx is convergent. Let

u = [nth square root of f (1/n) f(2/n)f(3/n) ... f(n/n).

Taking ln of both sides, we obtain

ln u = 1/n [[n.summation over (i=1) ln f (i/n)].

Therefore, by methods described before, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies that the required limit is equal to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We will record the result as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example 7 Using the above Example 6, the reader can show the following identities:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example 8 Evaluate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We will compute this limit using equation (4) with f( x) = x:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The limit in this example can also be calculated using the following command in Maple:

>limit(factorial(n)^(1/n)/n, n = infinity)

Though Maple uses sophisticated algorithms for finding limits, we cannot expect Maple to calculate any type of limit. In an attempt to calculate the limit

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in Example 4, the following Maple command does not produce an answer:

>limit((sum(k^m, k = 1 .. n))/n^(m+1), n = infinity)

In this paper, we have shown a method of calculating certain limits via suitable Riemann Sums. In some cases we were able to perform the calculation by hand, but in other cases, the calculation was facilitated by a CAS. However, we hope that the paper shed some light into the limitations of a CAS as illustrated by Example 4.

Acknowledgements

The author gratefully appreciates the editors and referees for their valuable comments.

References

[1] Norman Chonacky and David Winch. (2005). Reviews of Maple, Mathematica, and Matlab: Coming Soon to a Publication Near You. Computing in Science and Engineering, 7(2), 9-10.

[2] Yang, Hansheng and Heng Yang. (2001). The Arithmetic-Geometric Mean Inequality and the Constant e. Mathematics Magazine. 74(4): 321-323.

[3] George B. Thomas and Ross L. Finney. Calculus and Analytic Geometry (9th Edition) (Hardcover). Addison Wesley Longman (Higher Education Division, Pearson Education) . ISBN-10: 0201509008.

[4] Howard Anton and Albert Herr. Calculus With Analytic Geometry (Hardcover). John Wiley & Sons; 5 Sub edition ISBN-10: 0471594954.

[5] Sudhir K. Goel; Dennis M. Rodriguez.(1987). A Note on Evaluating Limits Using Riemann Sums. Mathematics Magazine, Vol. 60, No. 4, 225-228.

[6] Richard E. Klima; Neil Sigmon and Ernest Stitzinger. (2007). Applications of abstract algebra with Maple and MATLAB. Chapman andHall/CRC. ISBN: 1-58488-610-2.

[7] Henrik Aratyn and Constantin Rasinariu. (2006). A short course in mathematical methods with Maple. World Scientific Publishing Co. Pte. Ltd. ISBN:981-256-595-7.

Sanjay Kumar Khattri

sanjay.khattri@hsh.no

Department of Engineering

Stord Haugesund University College

5528

Norway

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Author: | Khattri, Sanjay Kumar |
---|---|

Publication: | Electronic Journal of Mathematics and Technology |

Article Type: | Report |

Geographic Code: | 4EXNO |

Date: | Oct 1, 2009 |

Words: | 1235 |

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