Mathematics 242 – Principles of Analysis Discussion 7 – Infinite Series November 20 and 27, 2000

Background

The last unit of this course will be devoted to the following general question: How can we tell whether an infinite sum of real numbers

(1)  ∑_{k=1}^{∞} a_{k}

“adds up” or converges to a well-defined finite result. A sum with infinitely many terms is called an infinite series. The numbers a_{k} are called the terms of the series.

We will approach this question by the following method. The thing that makes sense immediately from the algebraic properties of the real number system is to add a finite collection of numbers. So we can stop after any finite number N of terms in the infinite sum above and consider the “partial sum”:

s_{N} = ∑_{k=1}^{N} a_{k} = a_{1} + a_{2} + ⋯ + a_{N}

In this way, we get a sequence of partial sums, whose behavior should tell us whether the infinite sum (1) converges or not. Of course it is possible for an infinite sum not to add up to a finite result. For example, if a_{k} = 1 all k, then S_{N} = \frac{N(N+1)}{2} which goes to +∞ as N → ∞. We say that the series ∑_{k=1}^{∞} 1 diverges. On the other hand, if lim_{N→∞} s_{N} exists and equals S, then we will say that the series (1) converges to (intuitively, “adds up to”) S.

For today, we will concentrate on the case where all a_{k} ≥ 0.

Discussion Questions

A) Assuming all a_{k} ≥ 0, what can we say about the sequence of partial sums s_{N}? Under what conditions would that sequence be “guaranteed” to converge to some finite real value?

B) In problem 15 from Section 1.4 you proved that ∑_{k=1}^{N} \frac{1}{k!} ≤ 2 for all N. What does this say about the infinite series ∑_{k=1}^{∞} \frac{1}{k!}. Experiment with the numerical values for partial sums up to N = 15 and see if your statement seems reasonable.

C) In problem 11 from Section 1.4, you proved that if r is any real number, then the finite geometric series with ratio r adds up to the following:

∑_{k=0}^{N} r^{k} = \frac{1 − r^{N+1}}{1 − r} .

Suppose r > 0. Thinking of the left hand side here as the partial sum for the infinite geometric series ∑_{k=0}^{∞} r^{k}, when will it be true that this series converges?

D) Consider the series ∑_{k=1}^{∞} \frac{1}{k^{2}+k}, where a_{k} = \frac{1}{k^{2}+k} for all k ≥ 1. All partial sums of this series are rational numbers. Compute the partial sums s_{1}, …, s_{8} and look for a pattern. Prove your guess, and use that to determine an answer to our general question in this case. If ∑_{n=1}^{∞} \frac{1}{k^{2}+k} does add up to a finite result, what is the sum?

E) On the other hand, consider the series ∑_{k=1}^{∞} \frac{1}{k}. Consider the partial sums s_{2}, s_{4}, s_{8}, s_{16} and so forth–the partial sums where N = 2^{ℓ} is a power of 2. What can you say about s_{2^{ℓ+1}} −s_{2^{ℓ}}? Determine an answer to our general question in this case. If ∑_{n=1}^{∞} \frac{1}{k} does add up to a finite result, what is the sum?

F) Suppose you know that series ∑_{k=1}^{∞} a_{k} (all a_{k} ≥ 0) converges, and that the terms in a second series ∑_{k=1}^{∞} b_{k} satisfy b_{k} ≤ a_{k}. How are the Nth partial sums for the a-series and the b-series related? Must the b-series also converge? Prove your assertion.

G) Suppose you know that the ∑_{k=1}^{∞} a_{k} (all a_{k} ≥ 0) diverges, and that the terms in a second series ∑_{k=1}^{∞} b_{k} satisfy b_{k} ≥ a_{k}. How are the Nth partial sums for the a-series and the b-series related? Must the b-series also diverge? Prove your assertion.

H) Suppose you know that the ∑_{k=1}^{∞} a_{k} (all a_{k} ≥ 0) diverges, and that the terms in a second series ∑_{k=1}^{∞} b_{k} satisfy b_{k} ≤ a_{k}. Can you say anything in this case?

Assignment

Group write-ups due in class Wednesday, November 29.