We will cover following topics

Series And Its Convergence
Necessary and Sufficient Condition for Convergence
Absolute And Conditional Convergence Of Series
- Absolute Convergence
- Conditional Convergence
Rearrangement Of Series
PYQs
- Series And Its Convergence
- Rearrangement of Series

Series And Its Convergence

Any ordered infinite sequence of terms is called a series, and is denoted by $\sum_{i = 1}^{\infty} a_{i}$ .

A series $\sum_{n - 1}^{\infty} a_{n}$ converges if the sequence of the partial sums of the series converges.

The sequence of partial sums is defined as $(S_{1}, S_{2}, S_{3}, . . ., S_{n})$ , where $S_{n} = \sum_{k = 1}^{n} a_{k} \to l$

For example, $\sum_{n = 1}^{\infty} \frac{1}{n (n + 1)}$ converges because $S_{n} = 1 - \frac{1}{n + 1} \to 1$ . Similarly, for $0 < x < 1$ , the geometric series $\sum_{n = 0}^{\infty} x^{n}$ converges to $\frac{1}{1 - x}$ because $S_{n} = \frac{1 - x^{n + 1}}{1 - x}$ .

A necessary condition for $\sum_{n = 1}^{\infty} a_{n}$ to converge is that $a_{n} \to 0$ . For example, for $| x | \geq 1$ , $\sum_{n = 1}^{\infty} x^{n}$ diverges because $a_{n} ↛ 0$ . Similarly, $\sum_{n = 1}^{\infty} \sin n$ diverges because $a_{n} ↛ 0$ .

The above condition is necessary but not sufficient. For example, $\sum_{n = 1}^{\infty} \log (\frac{n + 1}{n})$ diverges, however, $\log (\frac{n + 1}{n}) \to 0$ .

Necessary and Sufficient Condition for Convergence

Theorem 1

Let $a_{n} \geq 0 \forall n$ . Then, $\sum_{n = 1}^{\infty} a_{n}$ converges if and only if the partial sum $S_{n}$ is bounded above.

For example, the harmonic series $\sum_{n = 1}^{\infty} \frac{1}{n}$ diverges because

S_{2^{k}} \geq 1 + \frac{1}{2} + 2 \cdot \frac{1}{4} + 4 \cdot \frac{1}{8} + \dots + 2^{k - 1} \cdot \frac{1}{2^{k}} = 1 + \frac{k}{2}

which diverges as $k \to \infty$ .

Theorem 2

If $\sum_{n = 1}^{\infty} | a_{n} |$ converges, then $\sum_{n = 1}^{\infty} a_{n}$ converges.

Comparison Test

If $0 \leq a_{n} \leq b_{n}$ for $n \geq k$ . Then,

(i) If $\sum_{n = 1}^{\infty} b_{n}$ converges $⟹ \sum_{n = 1}^{\infty} a_{n}$ converges.

For example, $\sum_{n = 1}^{\infty} \frac{1}{(n)^{2}}$ converges because $\frac{1}{(n) (n)} ⩽ \frac{1}{n (n - 1)}$ .

Here, $\frac{1}{n (n - 1)}$ is a difference series which converges to 1 as $n \to \infty$ . Therefore, $\sum_{n = 1}^{\infty} \frac{1}{n^{2}}$ converges.

Based on this result, we can also say that $\sum_{n = 1}^{\infty} \frac{1}{n!}$ converges because $\frac{1}{n!} < \frac{1}{n^{2}}$ for $n > 5$ .

(ii) If $\sum_{n = 1}^{\infty} a_{n}$ diverges, $⟹ \sum_{n = 1}^{\infty} b_{n}$ diverges.

For example, $\sum_{n = 1}^{\infty} \frac{1}{\sqrt{n}}$ diverges because $\frac{1}{n} ⩽ \frac{1}{\sqrt{n}}$ and the harmonic series $\sum_{n = 1}^{\infty} \frac{1}{n}$ diverges.

Limit Comparison Test

Let $a_{n}, b_{n} \geq 0$ and let $\frac{a_{n}}{b_{n}} \to L$ . Then,

If $L$ if positive and finite, then either both series converge or both series diverge.

Absolute And Conditional Convergence Of Series

Absolute Convergence

A series $\sum_{t = 0}^{\infty} a_{n}$ is said to converge absolutely if $\sum_{n = 0}^{\infty} | a_{n} | = L$ for some real number $L$ .

Conditional Convergence

A series $\sum_{n - 1}^{\infty} a_{n}$ converges conditionally if $\sum_{n - 1}^{\infty} a_{n}$ converges but the series $\sum_{n - 1}^{\infty} | a_{n} |$ diverges.

Rearrangement Of Series

Riemann Series Theorem: It says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series coverges to an arbitrary real number, or diverges.

PYQs

Series And Its Convergence

1) Find the range of $p (> 0)$ for which the series:

$\frac{1}{(1 + a)^{p}} - \frac{1}{(2 + a)^{p}} + \frac{1}{(3 + a)^{p}} - . . . ., a > 0$ , is

(i) absolutely convergent and (ii) conditionally convergent.

[2018, 10M]

Use comparison and Leibnitz tests.

2) Show that the series $\sum_{n = 1}^{\infty} \frac{(- 1)^{n + 1}}{n + 1}$ is conditionally convergent. (If you use any theorem, $(s)$ to show it, then you must give a proof of that theorem(s)).

[2016, 15M]

Use Leibnitz test

3) Test the convergence and absolute covergence of the series $\sum_{n = 1}^{\infty} (- 1)^{n + 1} (\frac{n}{n^{2} + 1})$ .

[2015, 10M]

Use Leibnitz test

4) Show that the series $\sum_{n = 1}^{\infty} {(\frac{π}{π + 1})}^{n} n^{6}$ is convergent.

[2012, 12M]

Use ratio test.

5) Show that the series ${(\frac{1}{3})}^{2}$ + ${(\frac{1.4}{3.6})}^{2}$ + $\dots . + {(\frac{1.4 .7 \dots . . (3 n - 2)}{3.6 .9 \dots \dots \dots . .3 n})}^{2}$ till infinity, converges.

[2009, 15M]

Use Rabee's test.

6) Find all the positive values of $a$ for which the series $\sum_{n = 1}^{\infty} \frac{(a n)^{n}}{n!}$ converges.

[2002, 12M]

Use ratio test.

Rearrangement of Series

1) Let $\sum_{n = 1}^{\infty} x_{n}$ be a conditionally convergent series of real numbers. Show that there is a rearrangement $\sum_{n = 1}^{\infty} x_{π (n)}$ of the series $\sum_{n = 1}^{\infty} x_{n}$ that converges to 100.

[2017, 20M]

2) Show that the series $\sum \frac{1}{n (n + 1)}$ is equivalent to $\frac{1}{2} \prod_{2}^{\infty} (1 + \frac{1}{n^{2} - 1})$ .

[2008, 15M]

3) Rearrange the series $\sum_{n = 1}^{\infty} (- 1)^{n + 1} \frac{1}{n}$ to converge to 1.

[2007, 20M]

< Previous Next >