I'm just starting to study limits of sequences and have been asked to determine if the series;
Sum (n=1 --> n = inf) 1/((n^2)+1) converges or diverges,
Is it correct for me to do the following,
Series will converge if the sequence (1/((n^2)+1) converges, hence;
take lim (n-->inf) (1/((n^2)+1)
and the limit = 0 as 1/inf is 0 in the limit.
?? is this correct? and should i be able to calculate the actual limit.
Cheers.
You seem to be saying that the series converges if and only if the sequence converges but that not true. If the sequence converges to any number other than 0, the series, does NOT converge. If the sequence converges to 0, the series may converge but not necessarily. for example 1/n converges to 0 but the series does NOT converge.
[/quote]take lim (n-->inf) (1/((n^2)+1)
and the limit = 0 as 1/inf is 0 in the limit.
?? is this correct? and should i be able to calculate the actual limit.
Cheers.[/QUOTE]
No, that is not correct. As simple pendulum says, you can use the comparison test to show that this particular series does converge. You could also use the integral test: is finite so this series converges. There are, in general, no good ways to actually find the sum.