Question

Proving that $\sum_{i=0}^{n-p} \frac{i!}{(p+i)!} = \frac{1}{p-1}[\frac{1}{(p-1)!}-\frac{(n-p+1)!}{n!}]$

score 95 · Answer 1 · 2026-04-17 20:30:14

95

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Proving that $\sum_{i=0}^{n-p} \frac{i!}{(p+i)!} = \frac{1}{p-1}[\frac{1}{(p-1)!}-\frac{(n-p+1)!}{n!}]$

Published on 17 Apr 2026 - 20:30

score 293 · Answer 2 · 2026-04-11 15:49:32

293

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Geometric Series Word Problem from Khan Academy

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85

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247

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score 153 · Answer 5 · 2026-04-13 16:26:23

153

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525

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score 105 · Answer 9 · 2026-04-12 23:09:42

How do we justify splitting a sum over $1 \le j < k+j \le n$ into two sums $1 \le k \le n$ and $1 \le j \le n-k$?

score 179 · Answer 10 · 2026-04-13 02:06:05

179

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score 4.6k · Answer 11 · 2026-04-15 04:35:55

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score 469 · Answer 12 · 2026-04-17 12:34:39

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score 287 · Answer 14 · 2026-04-14 06:20:26

287

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score 177 · Answer 15 · 2026-04-17 10:11:28

177

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List Question

Proving that $\sum_{i=0}^{n-p} \frac{i!}{(p+i)!} = \frac{1}{p-1}[\frac{1}{(p-1)!}-\frac{(n-p+1)!}{n!}]$

Geometric Series Word Problem from Khan Academy

Cauchy Product with reindexing

Changing the order summation and limit and proving a double-sequence identity

How find this sum $\sum_{k=0}^{n}\binom{n}{k}|n-2k|$ closed form or asymptotic behaviour?

Concrete Mathematics: How do we figure out the constrains of summations when using multiplication by summation factor method?

A Fourier series' upper bound involving gamma function

How to prove $\sum_{k=1}^{n}k\binom{n}{k} = n2^{n-1}$

How do we justify splitting a sum over $1 \le j < k+j \le n$ into two sums $1 \le k \le n$ and $1 \le j \le n-k$?

How to simplify $\sum_{a_1=1}^n\sum_{a_2=1}^{a_1}\sum_{a_3=1}^{a_2}\dots\sum_{a_{k+1}=1}^{a_k}1$

Sum of a decreasing geometric series of integers

Summation of the telescoping series $\sum_{n=1}^N \frac{x}{(1+(n-1)x)(1+nx)}$

Summation of a multiple series involving Fibonacci numbers

Sum of first $n$ integer values of a polynomial as a polynomial in $n$

Combinatoric Coefficients of a Polynomial

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