# Thread:

1. ## power series justification

Hi.

Quick question.
Given a power series of the form:

sigma cn(x+3)^n. It is given that this series converges when x=0 and diverges when
x = -8

Now does it converge when the series is

sigma cn2^n

The answer is yes but not sure how to justify.

2. Hint:

As $\sum_{n=0}^{+\infty}c_n3^n$ converges, the radius of convergence of the series $\sum_{n=0}^{+\infty}c_nx^n$ is $\geq 3$

3. Originally Posted by FernandoRevilla
Hint:

As $\sum_{n=0}^{+\infty}c_n3^n$ converges, the radius of convergence of the series $\sum_{n=0}^{+\infty}c_nx^n$ is $\geq 3$
I was trying to figure out the radius of convergence. This is what i have.

we know that |z-a| < R means that the series converges absolutely for some number R.
Here i have a as (-3) and z=x=0 when it converges.
so |-3| < R => |3| < R

now |3| can be + or - 3...so I guess is R > -3 or 3?
I dunno if im doing this right.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9084292054176331, "perplexity_flag": "middle"}