Convergence of Series

[K]

Suppose I have a series, say SUM(3/((2n + 1)^1/n)) from 1 to infinity.  Now we know that this series converges by p series; how can I find the value that the limit converges to?  This is all stuff I should know, but it's been so long.  


edit: suppose the series is SUM(1/eⁿ) 1 to infinity.  I like this better than the previous example.  This is a gemetric series, so my first thought was 1/(1 - e); this gives the correct answer only negative.  Hooh?

[Yoni]

You're wrong about the first series; it diverges to infinity.

About the second:
Sum [n=1 to infinity] ((1/e)^n)
Is an infinite converging geometric series.
The first element is: A1 = 1/e
The quotient is: q = 1/e
Therefore the sum is:
S = A1 / (1 - q) = (1/e) / (1 - 1/e) = 1 / e(1 - 1/e) = 1 / (e - 1)

[K]

You're wrong about the first series; it diverges to infinity.

Oops - you're right.  I got confused.  The sequence {3/((2n + 1)^1/n) } converges. (To 3, I think)

Edit: that means the series diverges by the Nth term test.

[Yoni]

Correct. Short proof:

Two basic limits are n^(1/n) -> 1 and c^(1/n) -> 1 for constant c > 0.
From this, by multiplying, you get, (2n)^(1/n) -> 1*1 = 1 and (3n)^(1/n) -> 1*1 = 1.
Using the Sandwich rule, you get (2n + 1)^(1/n) -> 1, therefore 3/((2n + 1)^(1/n)) -> 3/1 = 3.

This is also a proof that the matching series doesn't converge. For a series Sum(A(n)) to converge, it must satisfy A(n) -> 0. (Edit: You just said this in your edit. )