infinite series

[hottgye]

hey everyone im new here.

ok so i have this series and i want to find what values of x makes it converge, and also to find the some of the series for those values of x:


the sum (from n=0 to infinity) of: (4^n)(x^n)


i'm not really sure where to begin. i think you can consolidate the terms to make (4x)^n....but then what? could someone sort of explain what methods i could use and/or walk me through?


thanks so much guys

 

[stapel]

Ne'mind. Mis-read terms. Coulda swore there was a "/" in there. Sorry.

Eliz.

 

[pka]

Using the root test: [4<SUP>n</SUP>|x<SUP>n</SUP>|]<SUP>1/n</SUP>->4|x|<1.
Therefore, if |x|<(1/4) the series converges.
Clearly if x=1/4 or x=−1/4 the series diverges.

 

[soroban]

Hello, hottgye!

What values of x makes it converge?
Find the sum of the series for those values of x:

Sum (from n=0 to infinity) of: (4ⁿ)(xⁿ)
.

My favorite is the Ratio Test.
. . It doesn't always work, but it's the most powerful test IMHO.

. . . . . . . . . . . . . . . . . |a_n+1|
Form the ratio: .R .= .| ----- |
. . . . . . . . . . . . . . . . . | .a_n .|

. . . . . . . . . . . . . . .|4ⁿ⁺¹ xⁿ⁺¹|
We have: . R . = . |-------------| . = . |4x|
. . . . . . . . . . . . . . .| . 4ⁿ xⁿ . .|


If lim R < 1, the series converges.
. n->oo

So we have: . |4x| .< .1 . ---> . |x| < ¼

The series converges for: .-¼ < x < ¼

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Since |x| < ¼, then |4x| < 1.

The series is: .SUM (4x)ⁿ . = . 1 + 4x + (4x)² + (4x)³ + . . .

We have a geometric series with first term a = 1 and common ratio r = 4x

. . . . . . . . . . . . . . . . .1
The sum is: . S .= .--------
. . . . . . . . . . . . . . .1 - 4x