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## Hot answers tagged exponential-sum

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### What is the formula for finding the summation of an exponential function?

You can recognize your sum as a geometric sum which has the basic formula: $$\sum_{n=0}^N r^n = \frac{r^{N+1} - 1}{r-1}$$ To apply this to your sum $$\sum_{n=1}^{50} e^{-0.123(n)}$$ recognize that ...
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### All real number for which $n$ in $5^n+7^n+11^n=6^n+8^n+9^n$

Consider the function $f(x)=x^n$ for positive $x$. Its second derivative is $n(n-1)x^{n-2}$ and therefore for $n > 1$ or $n<0$ $f$ is strictly convex while for $0 < n < 1$ f is strictly ...
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### Standard way to evaluate $\sum_{k=0}^\infty \frac{x^k}{(k+1/2)!}$?

$$S(x)=\sum_{k=0}^\infty \frac{x^k}{(k+1/2)!}=\sum_{k=0}^\infty \frac{(2x)^k (2k)!!}{(2k+1)!}=\sum_{k=0}^\infty \frac{(4x)^k k!}{(2k+1)!}=\sum_{k=0}^\infty \frac{(4x)^k }{k!}\frac{(k!)^2}{(2k+1)!}$$ ...
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### exponential equation: $6^x+8^x+15^x=9^x+10^x+12^x$

Hint: your equation can be factorized as $$\left(2^x-3^x\right) \left(2^{2 x}+3^x-5^x\right)=0$$
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The typical way to "skip terms" in a power series is to use roots of unity. Here, let $\zeta=e^{i\pi/3}$ be a primitive $6$th root of unity, so that $$\frac16 \sum_{j=0}^5 (\zeta^\ell)^j = \begin{... • 73.6k 4 votes ### Prove the following series \sum\limits_{s=0}^\infty \frac{1}{(sn)!} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \... • 88.1k 4 votes ### Solving limits with ln without L’Hôpital’s rule Hint:$$ \ln(1+e^{ax})=\ln(e^{ax}(e^{-ax}+1))=ax+\ln(e^{-ax}+1)  If $a>0$, then $\lim_{x\to\infty}e^{-ax}=0$. However, you have to distinguishing between the cases $a>0$, $a=0$, $a<0$ and ...
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