http://www.ck12.org Chapter 8. Infinite Series
Theorem
If∑∞k= 1 ukconverges, then∑∞k= 1 uk+(u 1 +u 2 +...+um)is also convergent.
If∑∞k= 1 ukconverges, then∑∞k= 1 uk−(u 1 +u 2 +...+um)is also convergent.
Likewise, if∑∞k= 1 ukdiverges, then∑∞k= 1 uk+(u 1 +u 2 +...+um)and∑∞k= 1 uk−(u 1 +u 2 +...+um)are also divergent.
For a convergent series, adding or removing a finite number of terms will not affect convergence, but it will affect
the sum.
Example 12
Find the sum of∑∞k= 15 k^3 − 1 −( 3 +^35 ).
Solution
∑∞k= 15 k^3 − 1 is a geometric series witha=3 andr=^15. Its sum is^154
Then∑∞k= 15 k^3 − 1 −( 3 +^35 )=^154 −^185 = 203
Reindexing
Another property of convergent series is that we canreindexa series without changing its convergence. This means
we can start the indices of the series with another number other than 1. Keep the terms in order though for reindexing.
Example 13
∑∞k= 13 k^4 − 1 is a convergent geometric series. It can be reindexed by changing the starting position ofiand the power
ofi. The new series is still convergent.
∞
k∑= 14
3 k−^1 =∞
k∑= 64
3 k−^6You can check that the series on the right is the same series as the one of the left by writing out the first few terms
for each series. Notice that the terms are still in order.
Review Questions
- Express the number 111 as an infinite series.
- Finds 1 ,s 2 ,s 3 and for∑∞k= 1 (− 21 k)k.
- Determine if the infinite series 3+ 103 + 1032 + 1033 +...converges or diverges.
- What are the values ofaandrfor the geometric series 3+ 3 ( 2 )^1 + 3 ( 2 )^2 + 3 ( 2 )^3 +....?
Determine if each infinite series converges or diverges. If a series converges, find its sum.
5.∑∞k= 1 (^35 )k−^1
6.∑+k=∞ 1 (−^23 )k−^17.∑∞k= (^1) k 3 k−^35
8.∑∞k= 149 kk+−^21
- Find the sum of∑∞k= 2
((
−^23 )k−^1 + 5 k^1 − 1