find exact values for the summation
- Thread starter spdrmncoo
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pka
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\(\displaystyle \L \begin{array}{rcl}
e^x & = & \sum\limits_{n = 0}^\infty {\frac{{x^n }}{{n!}}} \\
xe^x & = & \sum\limits_{n = 0}^\infty {\frac{{x^{n + 1} }}{{n!}}} \\
e^x + xe^x & = & \sum\limits_{n = 0}^\infty {\frac{{\left( {n + 1} \right)x^n }}{{n!}}} \\
e^x + xe^x - 1 & = & \sum\limits_{n = 1}^\infty {\frac{{\left( {n + 1} \right)x^n }}{{n!}}} \\
x = 1& \quad & \Rightarrow \quad 2e - 1 = \sum\limits_{n = 1}^\infty {\frac{{\left( {n + 1} \right)x^n }}{{n!}}} \\
\end{array}\)
e^x & = & \sum\limits_{n = 0}^\infty {\frac{{x^n }}{{n!}}} \\
xe^x & = & \sum\limits_{n = 0}^\infty {\frac{{x^{n + 1} }}{{n!}}} \\
e^x + xe^x & = & \sum\limits_{n = 0}^\infty {\frac{{\left( {n + 1} \right)x^n }}{{n!}}} \\
e^x + xe^x - 1 & = & \sum\limits_{n = 1}^\infty {\frac{{\left( {n + 1} \right)x^n }}{{n!}}} \\
x = 1& \quad & \Rightarrow \quad 2e - 1 = \sum\limits_{n = 1}^\infty {\frac{{\left( {n + 1} \right)x^n }}{{n!}}} \\
\end{array}\)