Question

For what values of x are the following series convergent? In each case find the sum of the series for those values of x.
a) 1 + (x-4) + (x-4)^2 + (x-4)^3 +...
b) \[\sum_{n=1}^{\infty} 2^nx^n\]

Answer 1

What kind of test do you know to test for convergence?

Answer 2

I'm not sure what test, but i know this formula, S= a/1-r.

Answer 3

Okay, haven't done this in a while.. sorry. But I assume you mean the Geometric Series of Convergence Test?

Answer 4

http://prntscr.com/657xlc

Answer 5

yeah, it's geometric series

Answer 6

To test for the convergence just look at the value \(\color{red}{r}\)

Answer 7

If r is less than one, it converges.

Greater than or equal to, it diverges.

Answer 8

I'm talking about b btw, not a :P

Answer 9

I'm not sure how create the restrictions to make them converge.

Answer 10

I gtg, sorry :/

@ganeshie8 @Directrix @TheSmartOne @iambatman can help you :)

Answer 11

Well, you can say:
$$
\sum_{n=1}^\infty 2^n x^n = \sum_{n=1}^\infty (2x)^n = \sum_{n=1}^\infty u^n
$$And now say:
$$
u < 1 \implies 2x < 1 \implies x < \frac{1}{2} \\
\sum_{n=1}^\infty u^n = \frac{1}{1-u} = \frac{1}{1-2x}
$$

Answer 12

@pitamar

it should be

\(\mid u \mid <1\) in your solution

Answer 13

Yes I see it now, sorry =| @mathmath333

Answer 14

\(\large \begin{align} \color{black}{1 + (x-4) + (x-4)^2 + (x-4)^3 +\cdot \cdot \cdot \cdot \infty \hspace{.33em}\\~\\
(x-4)^0 + (x-4)^1 + (x-4)^2 + (x-4)^3 +\cdot \cdot \cdot \cdot \infty \hspace{.33em}\\~\\
\normalsize \text{let r=x-4} \hspace{.33em}\\~\\
(r)^0 + (r)^1 + (r)^2 + (r)^3 +\cdot \cdot \cdot \cdot \infty \hspace{.33em}\\~\\
\mid r \mid < 1 \hspace{.33em}\\~\\
\implies -1<r<1 \hspace{.33em}\\~\\
\implies -1<x-4<1 \hspace{.33em}\\~\\
\implies -1+4<x<1+4 \hspace{.33em}\\~\\
\implies 3<x<5 \hspace{.33em}\\~\\
}\end{align}\)