Find the sum of the series. $$ \sum_{n=0}^{\infty} \frac{(-1)^n 5^n x^{7n}}{n!} $$ We know that $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$. The series $ \sum_{n=0}^{\infty} \frac{(-1)^n 5^n x^{7n}}{n!} $ can be re-written as $ \sum_{n=0}^{\infty} \frac{(\text{_____})^n}{n!} $.
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Step 1: Recall the exponential series: e^t = sum_{n=0}^\infty t^n / n!. Show more…
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