Can anyone give me a conditionally convergent series, please?

Question

Other than the Alternating Harmonic Series which is:
[(-1)^(n-1)]/n

I need another one.

Answer 1

Leibniz criterion (see link below) for alternating series states that if a motonically decreasing sequence of positive real numbers (a_n) is such that a_n → 0 as n → ∞, then Σ (-1)^n * a_n is convergent. So, all you need do is take such a sequence where Σ a_n is divergent. Once such example is

1 + 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + ... + 1/p_n + ...

the sum of the reciprocals of the prime numbers (p_n); applying the Leibniz criterion yields that

1 - 1/2 + 1/3 - 1/5 + 1/7 - 1/11 + 1/13 - 1/17 + ...

is therefore a conditionally convergent series.

Answer 2

An important one is the expansion of (1 + x)^n when n is negative. This is only valid for -1 < x < 1.

Answer 3

(1/x)^n converges for x >= 2, diverges otherwise

Answer 4

(-2)^(-n)
or
(-1/n)^n