Summation Challenge: Evaluate $\sum_{n=0}^\infty \frac{16n^2+20n+7}{(4n+2)!}$

In summary, the Summation Challenge is a mathematical problem where the goal is to find the sum of an infinite series. The formula for this challenge is $\frac{16n^2+20n+7}{(4n+2)!}$, where n starts at 0 and goes to infinity. The "!" in the denominator represents the factorial function, and the challenge can be solved using various methods such as the ratio test, comparison test, or integration. The final answer is a decimal number that can be approximated but not found exactly.
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Evaluate $\displaystyle \sum_{n=0}^\infty \dfrac{16n^2+20n+7}{(4n+2)!}$.
 
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Consider the following two power series,

$\displaystyle \sin x=\sum_{n=0}^\infty (-1)^n\cdot \dfrac{x^{2n+1}}{(2n+1)!}$ and $\displaystyle e^x=\sum_{n=0}^\infty \dfrac{x^{n}}{n!}$, $x\in \mathbb{R} $

Hence we have

$\displaystyle \sin 1=\sum_{n=0}^\infty (-1)^n\cdot \dfrac{1}{(2n+1)!}=\sum_{n=0}^\infty \left(\dfrac{1}{(4n+1)!}-\dfrac{1}{(4n+3)!}\right)$ and

$\displaystyle e=\sum_{n=0}^\infty \dfrac{1}{n!}$

Using the above considerations we get

$\displaystyle \begin{align*} \sum_{n=0}^\infty \dfrac{16n^2+20n+7}{(4n+2)!}&=\sum_{n=0}^\infty \dfrac{(4n+2)(4n+1)+2(4n+2)+1}{(4n+2)!}\\&=\sum_{n=0}^\infty \left(\dfrac{1}{(4n)!}+\dfrac{2}{(4n+1)!}+\dfrac{1}{(4n+2)!}\right)\\&=\sum_{n=0}^\infty \dfrac{1}{n!}+\sum_{n=0}^\infty \left(\dfrac{1}{(4n+1)!}-\dfrac{1}{(4n+3)!}\right)\\&=e+\sin 1\end{align*}$
 

FAQ: Summation Challenge: Evaluate $\sum_{n=0}^\infty \frac{16n^2+20n+7}{(4n+2)!}$

1. What is the purpose of the summation challenge?

The purpose of the summation challenge is to evaluate the infinite series $\sum_{n=0}^\infty \frac{16n^2+20n+7}{(4n+2)!}$ and determine its value.

2. How is the summation challenge solved?

The summation challenge can be solved using various methods, such as the ratio test, the comparison test, or the integral test. These methods involve manipulating the given series and determining if it converges or diverges.

3. Why is the summation challenge important?

The summation challenge is important because it tests a person's understanding of infinite series and their ability to manipulate and evaluate them. It also has applications in various fields of science, such as physics, engineering, and mathematics.

4. What is the value of the summation challenge?

The value of the summation challenge is a real number that represents the sum of all the terms in the given infinite series. This value can be either finite or infinite, depending on whether the series converges or diverges.

5. What are some tips for solving the summation challenge?

Some tips for solving the summation challenge include understanding the properties of infinite series, using known tests and techniques, and being familiar with common series and their convergence or divergence. It is also helpful to break down the series into smaller parts and use algebraic manipulations to simplify it.

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