No harmonic, no foul

Calculus Level 4

n=1f(n)n\large \displaystyle\sum_{n=1}^{\infty} \dfrac{f(n)}{n}

Let SS denote the expression of the variable sum above where f(n)f(n) has a 50% chance of being 1 and a 50% chance of being -1 for each individual nn, (with the f(n)f(n)'s being independent of one another).

What is the probability that the series SS converges?

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