8.2. Infinite Series http://www.ck12.org
Example 5
Determine if the series 7+^78 + 872 + 873 +...+ 8 i^7 − 1 +...converges or diverges. If it converges, find the sum of the
series.
Solution
The series is a geometric series that can be written as∑∞k= 17 (^18 )k−^1. Thena=7 and the ratior=^18. Because∣∣^18 ∣∣< 1
, the series converges. The sum of the series is 1 −ar= 1 −^718 =^778 =8.
Example 6
Determine if the series∑+k=∞ 19 k−^1 converges or diverges. If it converges, find the sum of the series.
Solution
The series is a geometric series witha=1 and the ratior=9. Because| 9 |>1, the series diverges.
Example 7
Determine if^34 + 432 + 433 +...+^3 (− 4 k^1 )k+...converges or diverges. If it converges, find the sum of the series.
Solution
If we rewrite the series in terms of powers ofk, the series looks like this:
3 (− 1 )^1
41 +
3 (− 1 )^2
42 +
3 (− 1 )^3
43 +...+
3 (− 1 )k
4 k +...=^3(
−^14
) 1
+ 3
(
−^14
) 2
+...+ 3
(
−^14
)k
+....It looks like a geometric series witha=3 andr=−^14 .Since
∣∣
−^14
∣∣
=^14 <1, the series converges.
However, if we write the definition of a geometric series fora=3 andr=−^14 , the series looks like this:
+∞
k∑= 13(
−^14
k− 1 )
= 3(
−^14
) 0
+ 3
(
−^14
) 1
+ 3
(
−^14
) 2
+...
= 3 + 3
(
−^14
) 1
+ 3
(
−^14
) 2
+...
The original problem,^3 (− 411 )^1 +^3 (− 421 )^2 +^3 (− 431 )^3 +...+^3 (− 4 k^1 )k+..., does not have the leading term of 3. This does
not affect the convergence but will affect the sum of the series. We need to subtract 3 from the sum of the series
3 + 3 (−^14 )^1 + 3 (−^14 )^2 +...to get the sum of^3 (− 411 )^1 +^3 (− 421 )^2 +^3 (− 431 )^3 +...+^3 (− 4 k^1 )k+....
The sum of the series is: 1 −ar− 3 = 1 −(^3 − (^14) )− 3 =^354 − 3 =^125 − 3 =^125 −^155 =−^35.
Other Convergent Series
There are other infinite series that will converge.
Example 8
Determine if∑+k=∞ 1 (^2 k−k+^21 )converges or diverges. If it converges, find the sum.
Solution
Thenth partial sumsnis: