Math 162

Class 19

Infinite Series

Definition: Given an infinite sequence \(a = (a_n)_{n=k,\dots,\infty}\), we define the infinite series associated with \(a\) the formal sum \[a_k + a_{k+1} + a_{k+2} + \cdots = \sum_{i = k}^\infty a_i\] In other words, we place a \(+\) sign between each two consecutive terms of the sequence.

Question: Can this be interpreted as an actual sum, in other words, can we actually add all infinitely many terms of the sequence?

Example

\(\displaystyle a_n = \frac{1}{2^n}\) for \(n = 1, 2, dots\).

Sequence of Partial Sums

Given an infinite series \(\displaystyle\sum_{i = k}^\infty a_i = a_k + a_{k+1} + a_{k+2} + \cdots\), we define a sequence of partial sums of the series, \(s_n\) for \(n = k, k+1, \dots\) : \[s_n = \sum_{i = k}^n a_i = a_k + a_{k+1} + a_{k+2} + \cdots + a_n\]

If the sequence \(s_n\) converges, we say that the series \(\displaystyle \sum_{i = k}^\infty a_i\) is convergent, and we define \[\displaystyle \sum_{i = k}^\infty a_i = \lim_{n\to\infty} s_n\]

Otherwise, we say that the series diverges or is divergent.

Example

\(\displaystyle \sum_{i=1}^\infty\frac{1}{2^i}\)

Index shift

The following series are all the same:

\(\displaystyle \sum_{i=0}^\infty \frac{2^{n+3}}{n+5}\)
\(\displaystyle \sum_{i=1}^\infty \frac{2^{n+2}}{n+4}\)
\(\displaystyle \sum_{i=2}^\infty \frac{2^{n+1}}{n+3}\)
\(\displaystyle \sum_{i=3}^\infty \frac{2^{n}}{n+2}\)
\(\displaystyle \sum_{i=4}^\infty \frac{2^{n-1}}{n+1}\)

\(\displaystyle \sum_{i=5}^\infty \frac{2^{n-2}}{n}\)
\(\displaystyle \sum_{i=6}^\infty \frac{2^{n-3}}{n-1}\) \[\vdots\]

Examples

\(\displaystyle \sum_{i=1}^\infty 1\)
\(\displaystyle \sum_{i=1}^\infty (-1)^i\)
\(\displaystyle \sum_{i=1}^\infty \frac{i+1}{i}\)
\(\displaystyle \sum_{i=1}^\infty \frac{i}{i+1}\)

If \(\displaystyle\lim_{n\to\infty} a_n \neq 0\) then \(\displaystyle\sum_{i=k}^\infty a_i\) diverges.

This is known as the \(n\)-th term test for divergence.

Example (the harmonic series)

\(\displaystyle \sum_{i = 1}^\infty \frac{1}{n}\)

Trick: Look at the partial sums \(s_n\) where \(n\) is a power of 2:

Example

\(\displaystyle \sum_{i=1}^\infty \frac{1}{i^2 + i}\)

Important example (Geometric series)

\(\displaystyle \sum_{i=0}^\infty a\cdot r^i\)

Examples

\(\displaystyle\sum_{i=0}^\infty \left(\frac{2}{3}\right)^i\)
\(\displaystyle\sum_{i=1}^\infty \left(-\frac{2}{3}\right)^i\)
\(\displaystyle\sum_{i=0}^\infty \left(\frac{3}{2}\right)^i\)

Algebraic properties

Suppose \(\displaystyle\sum_{i=k}^\infty a_i\) and \(\displaystyle\sum_{i=k}^\infty b_i\) are both convergent.

Then:

\(\displaystyle\sum_{i=k}^\infty (a_i + b_i) = \sum_{i=k}^\infty a_i + \sum_{i=k}^\infty b_i\)
\(\displaystyle\sum_{i=k}^\infty (a_i - b_i) = \sum_{i=k}^\infty a_i - \sum_{i=k}^\infty b_i\)
For any real number \(c\), \(\displaystyle\sum_{i=k}^\infty c\cdot a_i = c\cdot\sum_{i=k}^\infty a_i\)

Example

\(\displaystyle \sum_{i = 1}^\infty \frac{1}{2^i} + \frac{7}{3^{i-1}}\)

Series with non-negative terms

Suppose \(a_n \ge 0\) for every \(n \ge k\). Then the sequence of partial sums of the series \(\displaystyle\sum_{i=k}^\infty a_i\) is increasing.

Then the series \(\displaystyle\sum_{i=k}^\infty a_i\) converges if and only if the sequence of its partial sums is bounded.

Example

\(\displaystyle\sum_{i=0}^\infty \frac{1}{2^i + 1}\)