Determine whether the series is convergent or divergent

[tnutty]

Homework Statement

Determine whether the series is convergent or divergent.

1 + 1/8 + 1/27 + 1/64 + 1/125 ...

Homework Equations

The Attempt at a Solution

I know this is convergent but not sure how to prove this mathematically.

 

[Feldoh]

It might help to write the series in it's summation form...

Do you know any of the tests for convergence?

 

[tnutty]

Well this chapter is about the integral test so can I apply that method?

 

[Feldoh]

Yeah you can apply that test, I suppose -- How does the integral test work?

 

[tnutty]

i don't think the integral test can apply to this because this cannot be represented as a function.

however, there is another test, the remainder test,

Remainder test states that :

Suppose f(k) = a_k, where f is a continuous,positive,decreasing function for x>=n
and $$\sum a_n$$ is convergent. If R_n= s-s_n,

then $$\int f(x) dx$$ $$\leq R_n$$ $$\leq \int f(x) dx$$, where the limit is from n to infinity

 

[Feldoh]

I think you're misunderstanding the integral test

It says that $$\sum_{n=1}^{infinity} f(n)$$ converges if $$\int_{n}^{infinity} f(x) dx$$ converges.

 

[tnutty]

Ok, I get that, but how can we represent the problem above as a sum of a series, so
we can use the integral test?

 

[Mark44]

Do you know what the general term (i.e., nth term) in your series looks like?

1 + 1/8 + 1/27 + 1/64 + 1/125 ... + ? + ...

There's a definite pattern going on here.

 

[tnutty]

Ya that's the problem, i can't seem to recognize any patterns.

 

[tnutty]

Oh man good eye, I just got off spring break and have been trying hard to get back into
the mode of thinking. Thanks man!.