[2024/05/07] Evaluate converging sum

Hint 1: ::: spoiler spoiler expand the expression :::

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Hint 2: ::: spoiler spoiler partial fractions :::

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Solution:

::: spoiler spoiler Link: https://gmtex.siri.sh/fs/1/School/Extra/Maths/Qotd%20solutions/2024-05-07_infinite-sum.html :::

Since this is everyone's favorite example of telescoping sums, let's do it another way just for giggles.

::: spoiler Combinatorial proof The denominator is P(n+1, 2) which is the number of ways for 2 specified horses to finish 1st and second in an n+1 horse race. So imagine you're racing against horses numbered {1, 2, 3, ....}. Either you win, which has probability 0 in the limit, or there is a lowest numbered horse, n, that finishes ahead of you. The probability that you beat horses {1,2, ... , n-1} but lose to n is (n-1)! / (n+1)! or P(n+1, 2) or 1/(n^2^+n), the nth term of the series. Summing these mutually exclusive cases exhausts all outcomes except the infinitesimal possibility that you win. Therefore the infinite sum is exactly 1.

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::: spoiler solution With partial fractions:

1/(n + n²) = 1/(n(n+1)) = A/n + B/(n+1)

A(n+1) + Bn = 1

n = 0 gives A = 1, n = -1 gives B = -1

1/(n+n²) = 1/n - 1/(n+1)

Σ (n = 1 to ∞) 1/(n+n²) = Σ (n = 1 to ∞) 1/n - Σ (n = 1 to ∞) 1/(n+1)

= Σ (n = 1 to ∞) 1/n - Σ (n = 2 to ∞) 1/n

= 1/1 + Σ (n = 2 to ∞) 1/n - Σ (n = 2 to ∞) 1/n

= 1

Guessing this is the standard solution :::