Q. 2 - Calculus

Question

A series can be thought of as an infinite sum $a_1+a_2+a_3+a_4+\cdots +a_k+\cdots$. A series converges if this sum gets closer and closer to some real number limit as we add up more and more terms. Otherwise, the series is said to diverge.

The series $1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\cdots +\frac{1}{k^2}+\cdots$ converges. Calculate partial sums including more and more terms until you are convinced that the sum eventually approaches a real number limit and does not grow without bound.

Although you might think that the series $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots +\frac{1}{k}+\cdots$ converges because its terms get smaller and smaller, you will see in Chapter 8 that it does not. Calculate partial sums including more and more terms until you are convinced that this sum diverges and in fact grows without bound, never approaching a real-number limit.

Step-by-Step Solution

Answer

The series $S=\frac{1}{1}+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+...+\frac{1}{k^2}$ should converge.

The series $S=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...+\frac{1}{k}$ should diverge.

Step 1. Given information.

Consider the given question,

$$\begin{gathered} S=\frac{1}{1}+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+...+\frac{1}{k^2} ...... \left(i\right) \\ S=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...+\frac{1}{k} ...... \left(ii\right) \end{gathered}$$

Step 2. Consider series (i).

Consider series (i),

When $$\begin{gathered} k=100, \\ S=1.634 \end{gathered}$$

When $$\begin{gathered} k=1000, \\ S=1.6439 \end{gathered}$$

When $$\begin{gathered} k=10000, \\ S=1.6448 \end{gathered}$$

Hence, one can say that with increasing value of 'k', the sum of series is increasing very slow and becoming nearly constant. Therefore, it should converge.

Step 2. Consider series (ii).

Consider series (ii),

When $$\begin{gathered} k=100, \\ S=5.18738 \end{gathered}$$

When $$\begin{gathered} k=1000, \\ S=7.48547 \end{gathered}$$

When $$\begin{gathered} k=10000, \\ S=9.78761 \end{gathered}$$

When $$\begin{gathered} k=1000000, \\ S=14.39272 \end{gathered}$$

When $$\begin{gathered} k=1000000000, \\ S=21.30048 \end{gathered}$$

Hence, one can say that with increasing value of 'k', the sum of series is increasing rapidly. Therefore, it should diverge.