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Results tagged with estimation
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user 13854
For questions about estimation and how and when to estimate correctly
4
votes
Estimating $\sum_{n=1}^\infty \frac{1}{(2n+1)^2}$ with integrals
Bounding with Integrals
Naively, we can bound with
$$
\frac16=\int_1^\infty\frac{\mathrm{d}x}{(2x+1)^2}\le\sum_{k=1}^\infty\frac1{(2n+1)^2}\le\int_0^\infty\frac{\mathrm{d}x}{(2x+1)^2}=\frac12
$$
Or w …
- 348k
4
votes
Mental estimate for tangent of an angle (from $0$ to $90$ degrees)
This may not be the best for in-your-head calculation, but for $x$ in degrees
$$
\tan\left(\frac{\pi x}{180}\right)\approx\frac{x(990-4x)}{(90-x)(630+4x)}\tag{1}
$$
is at most $0.6\%$ off for $0\le x\ …
- 348k
0
votes
Accepted
Lipschitz-type estimate... True or false?
Given any $\varepsilon\gt0$, let $x=\alpha^2+\epsilon$. Then $\frac{x-\alpha^2-\varepsilon}{\alpha}=0$ and $\frac{x-\alpha^2+\varepsilon}{\alpha}=\frac{2\varepsilon}\alpha$. Then
$$
\varphi\left(\frac …
- 348k
16
votes
Accepted
Is there a lower-bound version of the triangle inequality for more than two terms?
Note that
$$
|x+y|\ge{\large{|}}|x|-|y|{\large{|}}
$$
is a combination of
$$
|x|\le|y|+|x+y|\qquad\text{and}\qquad|y|\le|x|+|x+y|
$$
This same idea can be applied to the standard multi-term triangle …
- 348k
3
votes
Estimating the sum $\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$
I know the question is to show that the sum is greater than $2$, and Greg Martin's answer does that perfectly; however, I thought it might be interesting to compute the actual value of the sum.
Using …
- 348k