All Questions
Tagged with summation power-series
362
questions
1
vote
3
answers
145
views
Compute the following sum for any x?
Compute the following sum for any x?
$\sum_{n=0}^\infty {(x-1)^n\over (n+2)!}$
I am having trouble to compute that sum. It looks like geometric series but I don't know where to start.
Can everyone ...
- 111
0
votes
1
answer
67
views
differential equation using series expansion
Trying to solve xy'= xy + y using the series y(x) = $\sum\limits_{i=0}^\infty a_nx^n$
This is what i have so far.
y'(x)= $\sum\limits_{i=0}^\infty na_nx^{n-1}$
xy' - xy - y = 0
x $\sum\limits_{i=0}^...
- 87
1
vote
2
answers
5k
views
What is the general formula for power series summation? [duplicate]
While reviewing definite integrals, $\int_a^bf(x)dx$; I recalled that a definite integral could not only be solved by the difference of the anti-derivatives of intervals b and a, $F(b)-F(a)$, via the ...
- 138
8
votes
7
answers
446
views
Prove that $\sum\limits_{n=1}^\infty \frac{n^2(n-1)}{2^n} = 20$
This sum $\displaystyle \sum_{n=1}^\infty \frac{n^2(n-1)}{2^n} $showed up as I was computing the expected value of a random variable.
My calculator tells me that $\,\,\displaystyle \sum_{n=1}^\infty ...
- 35.9k
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
33
votes
3
answers
4k
views
Sum of Squares of Harmonic Numbers
Let $H_n$ be the $n^{th}$ harmonic number,
$$ H_n = \sum_{i=1}^{n} \frac{1}{i}
$$
Question: Calculate the following
$$\sum_{j=1}^{n} H_j^2.$$
I have attempted a generating function approach but ...
- 331
1
vote
1
answer
1k
views
Index of Summation Shift? Power Series and Differential Equations
I have never had to index shift a summation series before, and it seems relatively straightforward, however, I am looking at an example in my textbook that doesn't make sense. I am wondering if ...
- 175
3
votes
2
answers
144
views
Summation of infinite series
If we know the series sum given below converges to a value $C$(constant)
$$\sum_{n=0}^{\infty}a_n =C \tag 2$$ Can we generate following in terms of C. values of $a_n$ will tend to zero as n goes to ...
- 1,717
0
votes
3
answers
40
views
Finding the sum of a sequence of terms
$$1/1(2) - 1/3(2^3) + 1/5(2^5) - 1/7(2^7)$$
This is equal to
$$\sum_{n=0}^\infty(1/2)^{2n+1}(-1)^n/(2n+1)$$
Differentiating this leads to:
$$\sum_{n=0}^\infty(-1/4)^n$$
Which is equal to $4/5$
Thus, ...
- 323
0
votes
1
answer
39
views
Summation of infinte series
Sir,
I have three infinite summation
$A =J_1 \sum_{n=2}^\infty (n-1) f(n-2) \tag 1$ , $B =\sum_{n=0}^\infty f(n) \tag 2$ and $C =J_2\sum_{n=1}^\infty f(n-1) \tag 3$, with $f(0)=2,f(1)...
- 1,717
1
vote
2
answers
215
views
Matrix Inversion Test ( Sum of Matrix series)
Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. ,
Following are the given conditions
a) each $A_i$ is non invertible except $A_0$ because their determinant is zero.
b) $\...
- 1,717
7
votes
2
answers
732
views
Prove $(1-x)^{2k+1} \sum\limits_{n\ge 0}\binom{n+k-1}{k}\binom{n+k}{k} x^n = {\sum\limits_{j\ge 0} \binom{k-1}{j-1}\binom{k+1}{j} x^j} $
I stumbled upon the identity
$$(1-x)^{2k+1} \sum\limits_{n\ge 0}\binom{n+k-1}{k}\binom{n+k}{k} x^n = {\sum\limits_{j\ge 0} \binom{k-1}{j-1}\binom{k+1}{j} x^j}. $$
The right-hand side is a polynomial. ...
- 1,359
1
vote
1
answer
252
views
Hypergeometric function representation
Is it possible to express the following sum in terms of the hypergeometric function $_2F_1$:
$$
f(x) = \sum_{n=0}^\infty\frac{(-ax)^n}{n!~\Gamma(b-n)}
$$
with $a$ and $b$ constant values ($x>0$ ...
- 115
2
votes
0
answers
574
views
Abel's Theorem, alternate proof
I'm trying to solve:
Suppose $\sum_{n=1}^\infty a_n$ converges. Prove that:
$$
\lim_{r\to1^-}\sum_{n=1}^\infty r^n a_n = \sum_{n=1}^\infty a_n.
$$
Hint: Sum by parts.
In class, I have seen a ...
- 6,753
0
votes
2
answers
1k
views
Sum of potencies with higher potency as clue
I am supposed to calculate the following as simple as possible.
Calcute:
$$1 + 101 + 101^2 + 101^3 + 101^4 + 101^5 + 101^6 + 101^7$$
Tip: $$ 101^8 = 10828567056280801$$
I have absolutely no idea how ...
- 3