# Calculus

## Infinite Sequences and Series # Infinite Series

### Definitions

Let $$\left\{ {{a_n}} \right\}$$ be a sequence. Then the infinite sum

${\sum\limits_{n = 1}^\infty {{a_n}} }={ {a_1} + {a_2} + \ldots }+{ {a_n} + \ldots }$

is called an infinite series, or, simply, series. The partial sums of the series are given by

${\sum\limits_{n = 1}^n {{a_n}} }={ {a_1} + {a_2} + \ldots + {a_n},}$

where $${S_n}$$ is called the $$n$$th partial sum of the series. If the partial sums $$\left\{ {{S_n}} \right\}$$ converge to $$L$$ as $$n \to \infty,$$ then we say that the infinite series converges to $$L:$$

${\sum\limits_{n = 1}^\infty {{a_n}} = L,\;\;}\kern-0.3pt{\text{if}\;\;\lim\limits_{n \to \infty } {S_n} = L.}$

Otherwise we say that the series $$\sum\limits_{n = 1}^\infty {{a_n}}$$ diverges.

### $$N$$th term test

If the series $$\sum\limits_{n = 1}^\infty {{a_n}}$$ is convergent, then $$\lim\limits_{n \to \infty } {a_n} = 0.$$

#### Important!

The converse of this theorem is false. The convergence of $${{a_n}}$$ to zero does not imply that the series $$\sum\limits_{n = 1}^\infty {{a_n}}$$ converges. For example, the harmonic series $$\sum\limits_{n = 1}^\infty {\large\frac{1}{n}\normalsize}$$ diverges (see Example $$3$$), although $$\lim\limits_{n \to \infty } {a_n} = 0.$$

Equivalently, if $$\lim\limits_{n \to \infty } {a_n} \ne 0$$ or this limit does not exist, then the series $$\sum\limits_{n = 1}^\infty {{a_n}}$$ is divergent.

### Properties of Convergent Series

Let $$\sum\limits_{n = 1}^\infty {{a_n}} = A$$ and $$\sum\limits_{n = 1}^\infty {{b_n}} = B$$ be convergent series and let $$c$$ be a real number. Then

• $$\sum\limits_{n = 1}^\infty {\left( {{a_n} + {b_n}} \right)}$$ $$= A + B$$
• $$\sum\limits_{n = 1}^\infty {c{a_n}} = cA$$

## Solved Problems

Click or tap a problem to see the solution.

### Example 1

Determine whether $$\sum\limits_{n = 1}^\infty {\sqrt[\large n\normalsize]{3}}$$ converges or diverges.

### Example 2

Investigate convergence of the series $$\sum\limits_{n = 1}^\infty {\large\frac{{{e^n}}}{{{n^2}}}\normalsize}.$$

### Example 3

Show that the harmonic series $$\sum\limits_{n = 1}^\infty {\large\frac{1}{n}\normalsize}$$ diverges.

### Example 4

Investigate convergence of the series $$\sum\limits_{n = 0}^\infty {\left( {\large\frac{1}{{{3^n}}}\normalsize} + {\large\frac{1}{{{5^n}}}\normalsize} \right)}.$$

### Example 5

Investigate convergence of the series $$\sum\limits_{n = 1}^\infty {\large\frac{1}{{\left( {n + \pi } \right)\left( {n + \pi + 1} \right)}}\normalsize}.$$

### Example 6

Determine whether the series
${\frac{1}{{1 \cdot 2}} + \frac{1}{{2 \cdot 3}} }+{ \frac{1}{{3 \cdot 4}} + \frac{1}{{4 \cdot 5}} + \ldots }+{ \frac{1}{{n\left( {n + 1} \right)}} + \ldots }$
converges or diverges.

### Example 7

Evaluate $$\sum\limits_{n = 0}^\infty {\ln {\large\frac{{n + 2}}{{n + 1}}\normalsize}}.$$

### Example 1.

Determine whether $$\sum\limits_{n = 1}^\infty {\sqrt[\large n\normalsize]{3}}$$ converges or diverges.

Solution.

Since $$\lim\limits_{n \to \infty } \sqrt[\large n\normalsize]{3} = \lim\limits_{n \to \infty } {3^{\large\frac{1}{n}\normalsize}} = 1,$$ then the series $$\sum\limits_{n = 1}^\infty {\sqrt[\large n\normalsize]{3}}$$ diverges by the $$n$$th term test.

### Example 2.

Investigate convergence of the series $$\sum\limits_{n = 1}^\infty {\large\frac{{{e^n}}}{{{n^2}}}\normalsize}.$$

Solution.

Calculate the limit $$\lim\limits_{n \to \infty } {\large\frac{{{e^n}}}{{{n^2}}}\normalsize}.$$ Using L’Hopital’s rule, we find

${\lim\limits_{x \to \infty } \frac{{{e^x}}}{{{x^2}}} } = {\lim\limits_{x \to \infty } \frac{{{e^x}}}{{2x}} } = {\lim\limits_{x \to \infty } \frac{{{e^x}}}{2} = \infty .}$

Hence, the original series diverges by the $$n$$th term test.

Page 1
Problems 1-2
Page 2
Problems 3-7