calc infinite series question?

Question

1. (SUM) n=1 to n = infinity ln( n / 2n +1)

How would I find if this sumation notation function converges or
diverges. if converges, what would be the sum?

2. if (SUM) of A_n converges and A_n > 0 for all n, can anything be said about (SUM) 1 / A_n? what would be the reason behind it?

I need help getting answer to these problems. I'd appreciate it if you can help.

Thank you

Answer 1

ln(n/(2n+1) = ln(1/(2+(1/n)) -->ln(1/2)
is not alternating and is bounded below so must diverge

if sum a[n] converges and is not alternating then ultimately
a[n] < c/n for some constant
(if it was >= c/n then it would diverge becuase sum 1/n diverges)
therefore 1/a[n] > n/c
therefore sum 1/a[n] must diverge

Answer 2

The first one is <= sum(n=1 to infinity) of ln(n/2n)
= sum(n=1 to infinity) ln(1/2) = -infinity
so no question it diverges

2. No question at all the sum of 1/A_n diverges.
In order for the sum of A_n to converge, A_n -> 0, so 1/A_n -> infinity.
Therefore the sum certainly goes to infinity, and it goes pretty fast!

Answer 3

Using the ratio test, if the (n+1)th term divided by the nth term goes is less than 1 as n goes to infinity, then the summation converges. You can use this to test the first problem.

The second problem converges i.e. the sum of A_n.

Therefore A_n+1 / A_n < 1
What if we take the inverse series, B_n = 1/A_n
Then B_n+1 / B_n > 1, hence B_n does not converge.
The inverse series does not converge.

Answer 4

A necessary condition for convergence(by all means NOT SUFFICIENT) is lim A_n =0
1) lim A_n =ln(1/2) which is not zero.(all terms have the same sign: negative) so the series is divergent
2) If Sum A_n converges lim A_n=0
so lim 1/A_n is+ infinity and the series diverges

Answer 5

a geometrical sequence has a set ratio between consecutive words. an impression sequence does no longer. Restated, a geometrical sequence will be concept-about as particular case of an impression sequence the position the full sequence is prolonged by technique of a few consistent a and human being words have a coefficient of one million. Geometric: y = a? a million + x^2 + x^3 + x^4 + . . .x^n power: y = ? a + bx^2 + cx^3 + dx^4 + . . .?x^n, the position ? represents the nth coefficient.