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Results tagged with summation
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user 8955
Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.
7
votes
1
answer
302
views
Which numbers are sums of finite numbers of reciprocal squares?
Question: Is there a “nice” characterization of rational numbers $q$ for which $q$ can be written as
$$q = \frac{1}{n_1^2} + \frac{1}{n_2^2} + \dots + \frac{1}{n_k^2}$$
for distinct natural numbers $n …
- 10.6k
0
votes
1
answer
84
views
Why does $\sum_{n=0}^{\infty}{\frac{1}{n \cdot 2^n}} = \ln 2$? [duplicate]
The Wikipedia entry on $\ln 2$ includes this summation:
$$\sum_{n=0}^{\infty}{\frac{1}{n \cdot 2^n}} = \ln 2$$
I was very surprised when I saw this, and I don't have a clear sense of where it comes from …
- 10.6k
5
votes
The idea behind the sum of powers of 2
Here's a geometric intuition for why $2^0 + 2^1 + 2^2 + \dots + 2^n = 2^{n+1} - 1$:
Here's the idea. Notice that the boxes for $1 + 2 + 4$ are right next to a single box of size $8$. They perfectly f …
- 10.6k
6
votes
3
answers
340
views
Simplifying $\sum_{i=1}^n{\lfloor \frac{n}{i} \rfloor}$?
Is there a way to either get an exact value for this summation or find some simpler function it's asymptotically equivalent to? …
- 10.6k
5
votes
4
answers
205
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Simplifying or approximating $\sum_{k=1}^{\infty}\left(1 - \left(1 - 2^{-k}\right)^n\right)$?
Consider a game in which you flip a coin until you flip tails. Your score is then the number of heads you flipped. So, for example, the sequence $H$, $H$, $H$, $T$ has a score of three, while the sequ …
- 10.6k