Miscellaneous summations!

Algebra Level 4

∑ n = 1 ∞ 3 n 2 + n − 2 n ! = A e + B \large\ \sum _{n=1}^\infty \frac {3n^2 + n - 2}{n!} = Ae + B

The equation above holds true for some positive integers A A and B B . Find A B A^B .

Notation: e ≈ 2.718 e \approx 2.718 is the Euler's number .


The answer is 25.

Priyanshu Mishra by Priyanshu Mishra , 20, India

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2 solutions

S = ∑ n = 1 ∞ 3 n 2 + n − 2 n ! = ∑ n = 1 ∞ 3 n 2 − 3 n − 3 + 4 n + 1 n ! = ∑ n = 1 ∞ ( 3 n ( n − 1 ) ! − 3 ( n + 1 ) n ! + 4 ( n − 1 ) ! + 1 n ! ) = 3 0 ! + ∑ n = 0 ∞ 4 n ! + ∑ n = 0 ∞ 1 n ! − 1 0 ! = 2 + 5 e \begin{aligned} S & = \sum_{n=1}^\infty \frac {3n^2+n-2}{n!} \\ & = \sum_{n=1}^\infty \frac {3n^2-3n-3+4n+1}{n!} \\ & = \sum_{n=1}^\infty \left({\color{#3D99F6}\frac {3n}{(n-1)!} - \frac {3(n+1)}{n!}} + {\color{#D61F06}\frac 4{(n-1)!} + \frac 1{n!}} \right) \\ & = {\color{#3D99F6}\frac 3{0!}} + {\color{#D61F06} \sum_{n=0}^\infty \frac 4{n!} + \sum_{n=0}^\infty \frac 1{n!} -\frac 1{0!} } \\ & = 2 + 5e \end{aligned}

⟹ A B = 5 2 = 25 \implies A^B = 5^2 = \boxed{25}

5 Helpful 0 Interesting 0 Brilliant 0 Confused
Priyanshu Mishra
Dec 6, 2016

We observe that

S = ∑ n = 1 ∞ 3 n 2 + n − 2 n ! = ∑ n = 1 ∞ 3 n ( n − 1 ) + 4 n − 2 n ! = ∑ n = 1 ∞ ( 3 ( n − 2 ) ! + 4 ( n − 1 ) ! − 2 n ! ) = 5 e + 2 \begin{aligned} S & = \sum _{n=1}^\infty \frac {3n^2+n-2}{ n!} \\ & = \sum _{n=1}^\infty \frac {3n(n-1)+4n-2}{n!} \\ & = \sum _{n=1}^\infty \left( \frac 3{(n-2)!} +\frac 4{(n-1)!} -\frac 2{n!} \right) \\ & = 5e + 2 \end{aligned} .

The sum telescopes and we get A = 5 , B = 2 A=5, B= 2 , A B = 25 A^B = 25 .

3 Helpful 0 Interesting 0 Brilliant 0 Confused

You don't need to enter so many braces { }. It should be n = 1 n=1 to ∞ \infty instead of r = 1 r=1 to ∞ \infty . You can use "Toggle LaTex" in the pull-down menu ( ⋯ ) (\cdots) to see the codes.

Chew-Seong Cheong - 4 years, 6 months ago

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But how do you aligned that?

What was difference between my latex codes(which showed expressions here and there) and your latex code in my solution.?

Priyanshu Mishra - 4 years, 6 months ago

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Some of the braces { } affected the alignment. Don't use LaTex editor to enter. It enters the braces automatically. With some many of them it is difficult to check. For example you only need to key in x^2 instead of { x }^{ 2 }, \sum _ {k=1} ^\infty and \int _ 0^1 without the { } after them and \sin x instead of \sin {x}.

Chew-Seong Cheong - 4 years, 6 months ago

You need to use coprime integers in the problem question only when they are in a quotient A B \frac AB . Because if not there will be infinite solutions. It is standard to call e e as Euler's number in Brilliant and we should link it to our wiki.

Chew-Seong Cheong - 4 years, 6 months ago

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