Calculus

Infinite Sequences and Series

Infinite Series

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

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 on problem description to see 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