Find the sum of the series -nr-1r- r-

∑^n_r=1r× r!⇒∑^n_r=1r× r!+r! r! ⇒∑^n_r=1(r+1)r! r! ⇒∑^n_r=1(r+1)! r! ⇒ (n+1)! 1! = (n+1)! 1 ...

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Standard XII

Mathematics

Sum of n Terms

Find the sum ...

Question

Find the sum of the series $n \sum r = 1 r \times r!$

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Solution

$n \sum r = 1 r \times r! \Rightarrow n \sum r = 1 r \times r! + r! - r!$
$\Rightarrow n \sum r = 1 (r + 1) r! - r!$
$\Rightarrow n \sum r = 1 (r + 1)! - r!$
$\Rightarrow (n + 1)! - 1!$
$= (n + 1)! - 1$

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Find the sum of the series n∑r=1r×r!

n∑r=1r×r!⇒n∑r=1r×r!+r!−r!⇒n∑r=1(r+1)r!−r!⇒n∑r=1(r+1)!−r!⇒(n+1)!−1!=(n+1)!−1

Find the sum of the series $n \sum r = 1 r \times r!$

$n \sum r = 1 r \times r! \Rightarrow n \sum r = 1 r \times r! + r! - r!$
$\Rightarrow n \sum r = 1 (r + 1) r! - r!$
$\Rightarrow n \sum r = 1 (r + 1)! - r!$
$\Rightarrow (n + 1)! - 1!$
$= (n + 1)! - 1$