# Tag Info

Accepted

Accepted

### 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 ...

### 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 ...
Accepted

Accepted

### Standard way to evaluate $\sum_{k=0}^\infty \frac{x^k}{(k+1/2)!}$?

$$S(x)=\sum_{k=0}^\infty \frac{x^k}{(k+1/2)!}=\sum_{k=0}^\infty \frac{(2x)^k (2k)!!}{(2k+1)!}=\sum_{k=0}^\infty \frac{(4x)^k k!}{(2k+1)!}=\sum_{k=0}^\infty \frac{(4x)^k }{k!}\frac{(k!)^2}{(2k+1)!}$$ ...

### exponential equation: $6^x+8^x+15^x=9^x+10^x+12^x$

Hint: your equation can be factorized as $$\left(2^x-3^x\right) \left(2^{2 x}+3^x-5^x\right)=0$$