No harmonic, no foul

Thsi is known as a random harmonic series , which as a consequence of Kolmogorov's three-series theorem converges with probability $\boxed{1}.$

If you want to watch how it converges (generally to some number between -3 and 3 but can be more or less with low probability), this is a fun little program:

Python 2.7:

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import random
from fractions import Fraction as frac
from time import time
def f(n):
    return random.choice((1, -1))
print "How many seconds do you want to run simulations?"
seconds = int(raw_input("> "))
while True:
    total = 0
    n = 1
    end = time() + seconds
    while time() < end:
        total += frac(f(n),n)
        n += 1
        print float(total)
    print "Run again?"
    if raw_input("y/n: ") in ('y', ''):
        continue
    else:
        break

Let's write $X_n=f(n)/n$ . Then $\{X_n\}$ is a sequence of independent random variables, each with mean $0$ . Now, there is a result (which is essentially a version of Kolmogorov's three-series theorem) that states that $\sum_{n\geq 1} X_n$ converges with probability $1$ if $\sum_{n\geq 1} \text{Var}(X_n)<\infty$ . Here, we indeed have $\sum_{n\geq 1} \text{Var}(X_n)<\infty$ . Hence the given series converges with probability $1$ .