[2024/05/07] Evaluate converging sum
- Evaluate
SUM(1/(n + n^2))
from n = 1 to infty
SUM(1/(n + n^2))
from n = 1 to inftyzkfcfbzr, solutionWith 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
WhoresonWells, Since this is everyone’s favorite example of telescoping sums, let’s do it another way just for giggles.
Combinatorial proofThe 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. ___
siriusmart, (edited ) Hint 1:
spoilerexpand the expression
Hint 2:
spoilerpartial fractions
Solution:
spoilerLink: gmtex.siri.sh/fs/…/2024-05-07_infinite-sum.htmlhttps://lemmy.world/pictrs/image/b10bbfbe-728e-4873-ab6f-1758e61bbf13.png
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