CK-12-Calculus

http://www.ck12.org Chapter 8. Infinite Series

u 1 +u 2 +u 3 +u 4 +....

A shorthand notation for an infinite series is to use sigma notation:
∑∞k= 1 uk, which can be read as “the sum of the termsuk’s forkequal to 1 to infinity.”
We can make finite sums from the terms of the infinite series:

s 1 =u 1

s 2 =u 1 +u 2

s 3 =u 1 +u 2 +u 3

The first sum is the first term of the sequence. The second sum is the sum of the first two terms. The third term
is the sum of the first three terms. Thus, thenth finite sum,snis the sum of the firstnterms of the infinite series:
sn=u 1 +u 2 +u 3 +...+un.

Sequence of Partial Sums

As you can see, the sumssn=u 1 +u 2 +u 3 +...+unform a sequence. The sequence is very important for the study
of the related infinite series for it tells a lot about the infinite series.
Partial Sums
For an infinite series∑∞k= 1 uk, thenthpartial sum,snis the sum of the firstnterms of the infinite series:sn=∑nk= 1 uk
.
The sequence{sn}formed from these sums is called thesequence of partial sums.
Example 2
Find the first five partial sums of the infinite series 1+ 0. 1 + 0. 01 + 0. 001 +....
Solution

s 1 =u 1 = 1

s 2 =u 1 +u 2 = 1 + 0. 1 = 1. 1

s 3 = 1 + 0. 1 + 0. 01 = 1. 11

s 4 = 1 + 0. 1 + 0. 01 + 0. 001 + 0. 0001 = 1. 111

s 5 = 1 + 0. 1 + 0. 01 + 0. 001 + 0. 0001 = 1. 1111

To further explore series, try experimenting with this applet. The applet shows the terms of a series as well as
selected partial sums of the series. Series Applet. As you see from this applet, for some series the partial sums
appear to approach a fixed number, while for other series the partial sums do not. Exploring this phenomenon is the
topic of the next sections.

Convergence and Divergence

Just as with sequences, we can talk about convergence and divergence of infinite series. It turns out that the
convergence or divergence of an infinite series depends on the convergence or divergence of the sequence of partial
sums.