Determine whether series is convergent or divergent: 1/4 + 3/16 + 1/64 + 3/256 +...

[ThaddeusK]

Determine whether the series is convergent or divergent.

1/4 + 3/16 + 1/64 + 3/256 + 1/1024 + 3/4096 +...

If it is convergent, find the sum.

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This one's left me scratching my head. I can't think of an expression for the numerator that alternates between 1 and 3 that's not ridiculously complex.

Any help would be appreciated!


(Edited for better formatting)

 

[Deleted member 4993]

Determine whether the series is convergent or divergent.

1

4

+

3

16

+

1

64

+

3

256

+

1

1024

+

3

4096

+

If it is convergent, find the sum.

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This one's left me scratching my head. I can't think of an expression for the numerator that alternates between 1 and 3 that's not ridiculously complex.

Any help would be appreciated!

What if broke this down to two series:[1/4 + 1/64 + 1/1024.....] + 3*[1/16 + 3/256 + 3/4096 + ....]

=[1/4 + 1/64 + 1/1024.....] + 3/4 * [1/4 + 1/64 + 1/1024.....]

and continue.....

 

[ThaddeusK]

What if broke this down to two series:[1/4 + 1/64 + 1/1024.....] + 3*[1/16 + 3/256 + 3/4096 + ....]

=[1/4 + 1/64 + 1/1024.....] + 3/4 * [1/4 + 1/64 + 1/1024.....]

and continue.....

I think I see what you are saying, but I ran into another issue. Just to give some idea of my thought process, I'm trying to make the generalized expression fit sigma from 1 to infinity ar^(n-1) form, so I can evaluate the sum with the formula a/(1-r).


I thought of the expression 1/4^(2n-1), but the only way I can think to make the exponent n-1 is to have 1/4ⁿ * 1/4^(n-1). This doesn't seem right. Hopefully my explanation of what I'm trying is clear...

 

[stapel]

I think I see what you are saying, but I ran into another issue. Just to give some idea of my thought process, I'm trying to make the generalized expression fit sigma from 1 to infinity ar^(n-1) form, so I can evaluate the sum with the formula a/(1-r).

This is the form that the previous poster provided to you.

I thought of the expression 1/4^(2n-1), but the only way I can think to make the exponent n-1 is to have 1/4ⁿ * 1/4^(n-1). This doesn't seem right. Hopefully my explanation of what I'm trying is clear...

I'm sorry, but I don't follow...?

 

[Deleted member 4993]

What if broke this down to two series:[1/4 + 1/64 + 1/1024.....] + 3*[1/16 + 1/256 + 1/4096 + ....]

=[1/4 + 1/64 + 1/1024.....] + 3/4 * [1/4 + 1/64 + 1/1024.....]

and continue.....

=[1/4 + 1/64 + 1/1024.....] + 3/4 * [1/4 + 1/64 + 1/1024.....]

=(1 + 3/4) * 1/4 * [1 + 1/2⁴ + 1/2⁸.....]

=7/16 * 1/(1-1/16)

=7/15