Calculus

Infinite Sequences and Series

Infinite Sequences and Series Logo

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 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