How To Find Upper And Lower Bounds Of Series

Finding the upper and lower bounds of series is a fundamental concept in mathematical analysis, crucial for estimating the sum of infinite series when an exact solution is difficult or impossible to obtain. Understanding these bounds allows us to determine the convergence and approximation accuracy of series, which are pivotal in various fields like physics, engineering, and computer science.

Understanding Series and Bounds

A series is the sum of the terms of a sequence. Given a sequence ${a_n}_{n=1}^{\infty}$, the series associated with this sequence is the sum:

$ \sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + \dots $

The partial sum $S_n$ of the series is the sum of the first $n$ terms:

$ S_n = \sum_{k=1}^{n} a_k = a_1 + a_2 + \dots + a_n $

An upper bound of a series is a value that the sum of the series will never exceed. A lower bound is a value that the sum will never fall below. Determining these bounds is essential for understanding the behavior and convergence of a series.

Convergence and Divergence

Before diving into methods for finding upper and lower bounds, it's important to understand the concepts of convergence and divergence.

• Convergence: A series converges if its sequence of partial sums approaches a finite limit as $n$ goes to infinity:

  $ \lim_{n \to \infty} S_n = S $

  where $S$ is a finite number.

• Divergence: A series diverges if its sequence of partial sums does not approach a finite limit. This can happen in several ways: the partial sums might oscillate, tend to infinity, or have no discernible pattern.

Methods for Finding Upper and Lower Bounds

Several techniques can be used to find upper and lower bounds for series, depending on the properties of the series.

1. Comparison Test

The comparison test is a fundamental method for determining whether a series converges or diverges by comparing it to another series whose convergence properties are known. If we have a series $\sum a_n$ and we want to determine its behavior, we compare it to a series $\sum b_n$.

• Upper Bound:
  • If $0 \leq a_n \leq b_n$ for all $n$ greater than some $N$, and $\sum b_n$ converges, then $\sum a_n$ also converges. Moreover, the sum of $\sum b_n$ serves as an upper bound for the sum of $\sum a_n$.
  • Mathematically, if $\sum_{n=1}^{\infty} b_n = B$, then $B$ is an upper bound for $\sum_{n=1}^{\infty} a_n$.
• Lower Bound:
  • If $a_n \geq b_n \geq 0$ for all $n$ greater than some $N$, and $\sum b_n$ diverges, then $\sum a_n$ also diverges. However, the comparison test does not directly provide a lower bound. To find a lower bound, one might look at partial sums or other properties of the series.

Example:

Consider the series:

$ \sum_{n=1}^{\infty} \frac{1}{n^2 + 1} $

We can compare this to the convergent series:

$ \sum_{n=1}^{\infty} \frac{1}{n^2} $

Since $n^2 + 1 > n^2$, we have $\frac{1}{n^2 + 1} < \frac{1}{n^2}$ for all $n$. The series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ is a well-known convergent series (a p-series with $p = 2 > 1$), and its sum is $\frac{\pi^2}{6} \approx 1.64493$. Therefore, the sum of $\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}$ is less than $\frac{\pi^2}{6}$, making $\frac{\pi^2}{6}$ an upper bound.

To find a lower bound, we can look at the first few terms:

$ \sum_{n=1}^{\infty} \frac{1}{n^2 + 1} = \frac{1}{2} + \frac{1}{5} + \frac{1}{10} + \dots $

The sum of the first few terms can give us a reasonable lower bound. For instance, the sum of the first three terms is $\frac{1}{2} + \frac{1}{5} + \frac{1}{10} = 0.5 + 0.2 + 0.1 = 0.8$. This tells us that the sum of the series is at least 0.8.

2. Integral Test

The integral test provides a method for determining the convergence or divergence of an infinite series by comparing it to an improper integral.

Theorem:

If $f(x)$ is a continuous, positive, and decreasing function on the interval $[1, \infty)$, and $a_n = f(n)$ for all integers $n \geq 1$, then the series $\sum_{n=1}^{\infty} a_n$ converges if and only if the improper integral $\int_{1}^{\infty} f(x) , dx$ converges.

• Upper Bound:
  • If $\int_{1}^{\infty} f(x) , dx$ converges, the value of the integral gives an upper bound for the series $\sum_{n=1}^{\infty} a_n$, but typically adjusted by adding the first term $a_1$. The upper bound $U$ can be expressed as: $ U = a_1 + \int_{1}^{\infty} f(x) , dx $
• Lower Bound:
  • A lower bound can be found by looking at the integral from 0 to infinity, assuming the function is well-defined there, or by summing the initial terms up to some point and then integrating from that point to infinity. If we consider the integral from $2$ to $\infty$, we get a lower bound by adding the first term $a_1$: $ L = \sum_{n=1}^{N} a_n + \int_{N+1}^{\infty} f(x) , dx $

Example:

Consider the series:

$ \sum_{n=1}^{\infty} \frac{1}{n^2} $

The function $f(x) = \frac{1}{x^2}$ is continuous, positive, and decreasing on $[1, \infty)$. We can evaluate the integral:

$ \int_{1}^{\infty} \frac{1}{x^2} , dx = \lim_{t \to \infty} \int_{1}^{t} \frac{1}{x^2} , dx = \lim_{t \to \infty} \left[-\frac{1}{x}\right]{1}^{t} = \lim{t \to \infty} \left(-\frac{1}{t} + 1\right) = 1 $

Thus, $\int_{1}^{\infty} \frac{1}{x^2} , dx = 1$.

• Upper Bound: The upper bound is $a_1 + \int_{1}^{\infty} f(x) , dx = 1 + 1 = 2$.
• Lower Bound: We know that $\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \approx 1.64493$. We can confirm that this lies between our bounds of 1 and 2.

3. Ratio Test

The ratio test is useful for determining the convergence or divergence of a series by examining the ratio of consecutive terms.

Given a series $\sum a_n$, compute the limit:

$ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| $

• If $L < 1$, the series converges.
• If $L > 1$, the series diverges.
• If $L = 1$, the test is inconclusive.

The ratio test primarily helps in determining convergence or divergence but does not directly provide upper and lower bounds. However, it can be used in conjunction with other methods to estimate the bounds.

4. Alternating Series Test (Leibniz's Test)

The alternating series test is applicable to series of the form:

$ \sum_{n=1}^{\infty} (-1)^{n-1} b_n = b_1 - b_2 + b_3 - b_4 + \dots $

where $b_n > 0$ for all $n$.

Theorem:

If the sequence ${b_n}$ is decreasing and $\lim_{n \to \infty} b_n = 0$, then the alternating series converges.

• Upper and Lower Bounds: The partial sums of an alternating series oscillate around the true sum. The sum $S$ lies between any two consecutive partial sums.

  • If $S_n$ is the $n$-th partial sum, then $S_{2n} < S < S_{2n+1}$ for even partial sums and $S_{2n+1} > S > S_{2n}$ for odd partial sums.

  Therefore, consecutive partial sums provide increasingly accurate upper and lower bounds for the sum of the series.

Example:

Consider the alternating series:

$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots $

Here, $b_n = \frac{1}{n}$, which is decreasing and approaches 0 as $n \to \infty$. Thus, the series converges.

Let's find some partial sums:

• $S_1 = 1$
• $S_2 = 1 - \frac{1}{2} = \frac{1}{2} = 0.5$
• $S_3 = 1 - \frac{1}{2} + \frac{1}{3} = \frac{5}{6} \approx 0.8333$
• $S_4 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} = \frac{7}{12} \approx 0.5833$

We have:

• $S_2 < S < S_3 \implies 0.5 < S < 0.8333$
• $S_4 < S < S_5 \implies 0.5833 < S < S_5$

The true sum of this series is $\ln(2) \approx 0.6931$, which lies between these bounds.

5. Telescoping Series

A telescoping series is one in which consecutive terms cancel each other, simplifying the partial sums.

General Form:

$ \sum_{n=1}^{\infty} (b_n - b_{n+1}) $

The partial sum $S_n$ is:

$ S_n = (b_1 - b_2) + (b_2 - b_3) + (b_3 - b_4) + \dots + (b_n - b_{n+1}) = b_1 - b_{n+1} $

If $\lim_{n \to \infty} b_{n+1} = L$ exists, then the series converges to $b_1 - L$.

• Upper and Lower Bounds: In this case, the sum $S = b_1 - L$. If $b_n$ is decreasing, then $b_1 - b_2$ is a lower bound, and $b_1$ is an upper bound. Conversely, if $b_n$ is increasing, $b_1 - b_2$ is an upper bound and $b_1$ is a lower bound, but $b_1 - L$ is still the value to which the series converges.

Example:

Consider the series:

$ \sum_{n=1}^{\infty} \frac{1}{n(n+1)} $

We can rewrite this as:

$ \sum_{n=1}^{\infty} \left(\frac{1}{n} - \frac{1}{n+1}\right) $

The partial sum $S_n$ is:

$ S_n = \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \dots + \left(\frac{1}{n} - \frac{1}{n+1}\right) = 1 - \frac{1}{n+1} $

As $n \to \infty$, $\frac{1}{n+1} \to 0$, so the series converges to 1.

• Upper and Lower Bounds: Since the series converges to 1, 1 is both the upper and lower bound in the limit. However, for any finite $n$, the partial sum is always less than 1, so 1 is always an upper bound.

6. Bounding with Known Series

Sometimes, we can use known series to bound a more complex series. Common series to use for comparison include geometric series, p-series, and exponential series.

Example:

Consider the series:

$ \sum_{n=1}^{\infty} \frac{2^n + 1}{3^n} $

We can split this into two series:

$ \sum_{n=1}^{\infty} \frac{2^n}{3^n} + \sum_{n=1}^{\infty} \frac{1}{3^n} = \sum_{n=1}^{\infty} \left(\frac{2}{3}\right)^n + \sum_{n=1}^{\infty} \left(\frac{1}{3}\right)^n $

Both are geometric series. The first one converges to:

$ \sum_{n=1}^{\infty} \left(\frac{2}{3}\right)^n = \frac{\frac{2}{3}}{1 - \frac{2}{3}} = \frac{\frac{2}{3}}{\frac{1}{3}} = 2 $

And the second one converges to:

$ \sum_{n=1}^{\infty} \left(\frac{1}{3}\right)^n = \frac{\frac{1}{3}}{1 - \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2} $

Therefore, the original series converges to $2 + \frac{1}{2} = \frac{5}{2} = 2.5$.

• Upper and Lower Bounds: Since the sum is 2.5, any partial sum will be a lower bound, and 2.5 is the ultimate upper bound as $n \to \infty$.

Practical Considerations

• Computational Tools: Software like Mathematica, Maple, or Python with libraries like NumPy and SciPy can be used to compute partial sums and approximate integrals, aiding in the estimation of bounds.
• Error Estimation: Understanding the rate of convergence is crucial for estimating the error when using partial sums as approximations. Techniques like the remainder estimation for the integral test or alternating series test can provide error bounds.
• Numerical Analysis: When analytical methods are insufficient, numerical analysis techniques such as Richardson extrapolation can be used to improve the accuracy of approximations.

Conclusion

Finding upper and lower bounds of series is a critical skill in mathematical analysis. By understanding various methods like the comparison test, integral test, ratio test, alternating series test, and telescoping series, we can effectively estimate the sums of infinite series and determine their convergence properties. The choice of method depends on the characteristics of the series. Combining these techniques with computational tools and error estimation methods allows for precise and practical approximations in various scientific and engineering applications. Properly bounding series is crucial not only for theoretical analysis but also for practical problem-solving, where accuracy and reliability are paramount.