A problem on the basics of convergence of a series

I was taught that if a series converge ,

for an example if {an} converges then lim ∑an = 0

But again I was taught that , an = (-1)^n is absolutely convergent and it converges to 1.

I think it contradicts the prior theory , because it does not equal to 0.

I think I'm missing a theory part , can someone please elucidate ?

thank you so much!

• The theorem should be: $$\text{If }\sum_{n=0}^{\infty}a_n\text{ is a convergent series, then }a_n\to0$$ Neither the sequence $a_n=(-1)^n$ nor the series $\sum_{n=0}^{\infty}(-1)^n$ are convergent. – logarithm May 18 at 13:55
• If the series converges the sum is not necessarily $0$. The sum is the limit of the sequence of partial sums, whatever that turns out to be. You were certainly not taught that $a_n=(-1)^n$ is a convergent series, absolutely or conditionally. – saulspatz May 18 at 13:56
• Both things you say you were taught are obviously wrong. You need to be much more careful - you're totally scrambling things. – David C. Ullrich May 18 at 13:59
• No sir, she taught me that it is absolutely convergent , when n is odd it is -1 and when n is even it is +1 thus |(-1)^n| is +1 , that is what she taught me @saulspatz – lasan manujitha May 18 at 18:17