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Results tagged with sequences-and-series
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user 164587
For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.
5
votes
How to show that $\sum_{n=1}^{\infty} \frac{1}{n^k}$ converges for all integer $k > 1$?
Compare $\frac{1}{n^2}$ with $\frac{1}{n(n-1)}$ ($n \ge 2$).
The second of these decomposes into partial fractions and the infinite sum can be easily computed.
- 1,509
2
votes
Accepted
Increasing sequence and converging to zero.
To prove $x_n$ is increasing:
\begin{align*}
x_n &= \sqrt{x_{n+1}-1} \implies x_{n+1} = (x_n+1)^2 - 1 = x_n^2 + 2x_n \\
\implies x_{n+1} - x_n &= x_n(x_n+1) > 0 \qquad \text{where $-1 \lt x_n \lt 0$} …
- 1,509