Difference between revisions of "009C Sample Final 3, Problem 6"

Revision as of 11:57, 18 February 2017

Consider the power series

\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{n+1}}{n+1}}

a) Find the radius of convergence of the above power series.

b) Find the interval of convergence of the above power series.

c) Find the closed formula for the function

f(x)

to which the power series converges.

d) Does the series

\sum _{n=0}^{\infty }{\frac {1}{(n+1)3^{n+1}}}

converge? If so, find its sum.

Solution:

(a)

Step 2:

(b)

Step 2:

Final Answer:
(a)
(b)

@@ Line 1: / Line 1: @@
-<span class="exam">Compute
+<span class="exam"> Consider the power series
-::<span class="exam">a) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{3-2n^2}{5n^2+n+1}</math>
+::::<math>\sum_{n=0}^\infty (-1)^n \frac{x^{n+1}}{n+1}</math>
-::<span class="exam">b) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{\ln n}{\ln 3n}</math>
+::<span class="exam">a) Find the radius of convergence of the above power series.
+::<span class="exam">b) Find the interval of convergence of the above power series.
+::<span class="exam">c) Find the closed formula for the function <math>f(x)</math> to which the power series converges.
+::<span class="exam">d) Does the series
+::::<math>\sum_{n=0}^\infty \frac{1}{(n+1)3^{n+1}}</math>
+::<span class="exam">converge? If so, find its sum.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"