All Questions
10
questions
6
votes
2
answers
115
views
Proving that the exponential satisfies the following sum equation
I was thinking about how $(\sum_{n=0}^{\infty} \frac{1}{n!})^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}
$
and was wondering if there existed any other sequences that satisfied this besides the exponential....
- 506
0
votes
2
answers
119
views
How do I find the partial sum of the Maclaurin series for $e^x$?
In one of the problems I am trying to solve, it basically narrowed down to finding the sum $$\sum^{n=c}_{n=0}\frac{x^n}{n!}$$ which is the partial sum of the Maclaurin series for $e^x$.
Wolfram | ...
- 3,515
1
vote
3
answers
131
views
How to prove that $\sum _{n=0}^{\infty }\:\frac{(x^n)'}{(n-1)!} = e^{x}(x-1)$
I am trying to prove that $$\sum _{n=0}^{\infty }\:\frac{\left(x^n\right)'}{\left(n-1\right)!} = e^{x}(x+1)\tag 1$$
This sum is very similar to the derivative of exponential $(e^x)' = \sum _{n=0}^{\...
- 23
1
vote
1
answer
49
views
Exponent-like power series with coefficients of increasing complexity
How one deals with series like this one:
$zb + \frac{z^2}{2!}b^2(1+\frac{c}{b}) + \frac{z^3}{3!}b^3(1+\frac{2c}{b})(1+\frac{c}{b}) + \frac{z^4}{4!}b^4(1+\frac{3c}{b})(1+\frac{2c}{b})(1+\frac{c}{b})+......
- 63
0
votes
4
answers
74
views
Show that $\sum_{n=1}^{\infty} nx^n/(n-1)! = e^xx(x+1)$
Please excuse my relatively novice skills, I'm first year (of 5) on my masters in mathematics.
I'm trying to show that
$$
\sum_{n=1}^{\infty} \frac{nx^n}{(n-1)!} = e^xx(x+1), \forall x.
$$
I already ...
- 19
0
votes
1
answer
1k
views
Summation of infinite exponential series
How is the given summation containing exponential function
$\sum_{a=0}^{\infty} \frac{a+2} {2(a+1)} X \frac{(a+1){(\lambda X)}^a e^{-\lambda X}}{a! (1+\lambda X)}=\frac{X}{2} (1+ \frac{1}{1+\lambda X})...
- 93
4
votes
2
answers
401
views
Infinite sum involving powers and factorials
I am interested in evaluating the following infinite sum
\begin{equation}
\sum_{n=0}^{\infty} \frac{\alpha^{n}}{n!}n^{\beta}
\end{equation}
where both $\alpha$ and $\beta$ are real numbers. However, ...
- 561
-1
votes
1
answer
2k
views
Finite power series [duplicate]
I'm a student and I'm looking for a solution for the following finite power series:
$$
\sum_{n=0}^m \frac{1}{n!} x^n
$$
By "solution" I meant expansion of the series and finding a closed form ...
- 113
4
votes
1
answer
4k
views
Exponential series is cosh(x), how to show using summation?
I want to show that $$\cosh(x) = \sum_{n=0}^{\infty} \frac{(x)^{2n}}{(2n)!}
$$
I know that $cosh(x) = \frac{exp(x)+exp(-x)}{2}$ but i cant seem to get there from the original series. I know that $$...
- 635
1
vote
2
answers
3k
views
Proof that sum of power series equals exponential function?
I have found that the Sum series equal an exponential function as below, however I have not found a proof for it:
$$
ze^z = \sum_{k=0}^{\infty} k \frac{z^k}{k!}
$$
I have though managed to prove ...
- 69