1+(-1)n/n, Any Idea?

The sum over n=1..Infinity of the series whose terms are (x^n)/n is -ln[1-x].

Your series is 1+sum[n=1..Infinity, ((-1)^n)/n] = 1 - ln[1-(-1)] = 1-ln[2], approximately 0.30685. Thus, the sum of the terms you defined is definitely convergent.

ln[x] is the natural logarithm of x, but you know that.

=1+(-1)n/n
=1-1
=0

1+(-1)n/n

If we put n=infinity then the result is 1. So the sequence
converges and its limit is 1.