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Question Number 47119 by math solver by abdo. last updated on 05/Nov/18

l e t u_{n} = \int_{- \infty}^{\infty} e^{- {n x}^{2} + x} d x

1) c a l c u l a t e u_{n}

2) f i n d \sum_{n} u_{n}

Commented by maxmathsup by imad last updated on 05/Nov/18

1) w e h a v e u_{n} = \int_{- \infty}^{+ \infty} e^{- {{(\sqrt{n} x)}^{2} - 2 x \frac{1}{2 \sqrt{n}} + \frac{1}{4 n} - \frac{1}{4 n}}} d x

= e^{\frac{1}{4 n}} \int_{- \infty}^{+ \infty} e^{- {(\sqrt{n} x - \frac{1}{\sqrt{n}})}^{2}} d x =_{\sqrt{n} x - \frac{1}{\sqrt{n}} = t} e^{\frac{1}{4 n}} \int_{- \infty}^{+ \infty} e^{- t^{2}} \frac{d t}{\sqrt{n}}

= \frac{e^{\frac{1}{4 n}}}{\sqrt{n}} \sqrt{π} \Rightarrow u_{n} = \frac{\sqrt{π}}{\sqrt{n}} e^{\frac{1}{4 n}} .