# A Series Problem!

Dear friends i am having a problem in series with factorials

$$\displaystyle \sum _{ r=1 }^{n }{ \frac { { r }^{ 2 }-r-1 }{ (r+1)! } }$$

i need to know how to solve the above problem. a solution would be welcome.

Note by Nishant Singh
2 years, 8 months ago

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Do it like this:$\sum _{ r=1 }^{ n }{ \frac { { r }^{ 2 }-r-1 }{ \left( r+1 \right) ! } } \\ =\sum _{ r=1 }^{ n }{ \frac { \left( { r }^{ 2 }+r \right) -2\left( r+1 \right) +1 }{ \left( r+1 \right) ! } } \\ =\sum _{ r=1 }^{ n }{ \left\{ \frac { \left( { r }^{ 2 }+r \right) }{ \left( r+1 \right) ! } -\frac { 2\left( r+1 \right) }{ \left( r+1 \right) ! } +\frac { 1 }{ \left( r+1 \right) ! } \right\} } \\ =\sum _{ r=1 }^{ n }{ \frac { \left( { r }^{ 2 }+r \right) }{ \left( r+1 \right) ! } } -2\sum _{ r=1 }^{ n }{ \frac { \left( r+1 \right) }{ \left( r+1 \right) ! } } +\sum _{ r=1 }^{ n }{ \frac { 1 }{ \left( r+1 \right) ! } } \\ =\sum _{ r=1 }^{ n }{ \frac { 1 }{ \left( r-1 \right) ! } } -2\sum _{ r=1 }^{ n }{ \frac { 1 }{ r! } } +\sum _{ r=1 }^{ n }{ \frac { 1 }{ \left( r+1 \right) ! } } \\ =\sum _{ x=0 }^{ n-1 }{ \frac { 1 }{ x! } } -2\sum _{ x=0 }^{ n-1 }{ \frac { 1 }{ \left( x+1 \right) ! } } +\sum _{ x=0 }^{ n-1 }{ \frac { 1 }{ \left( x+2 \right) ! } }\\=\left( \frac { 1 }{ 0! } +\frac { 1 }{ 1! } +\frac { 1 }{ 2! } +...+\frac { 1 }{ \left( n-1 \right) ! } \right) -2\left( \frac { 1 }{ 1! } +\frac { 1 }{ 2! } +...+\frac { 1 }{ n! } \right) +\left( \frac { 1 }{ 2! } +\frac { 1 }{ 3! } +...+\frac { 1 }{ \left( n+1 \right) ! } \right) \\ =\frac { 1 }{ 0! } -\frac { 1 }{ 1! } -\frac { 1 }{ n! } +\frac { 1 }{ \left( n+1 \right) ! } \\ =-\frac { n }{ \left( n+1 \right) ! }$

- 2 years, 6 months ago

Thank You! for the solution

- 2 years, 6 months ago

You're welcome! Hope I could make it clear..

- 2 years, 6 months ago

Hint:- Try to find a telescoping series.

- 2 years, 8 months ago

Tried it but I'm unable to split in partial fractions.

- 2 years, 8 months ago

You should be getting partial fractions of the form $$\frac{r}{(r+1)!}$$.

- 2 years, 8 months ago