# Tagged Questions

Questions on the evaluation and properties of limits.

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### Power Quantum Series And It's Sum.

Let $I$ be the interval $(-\theta, \theta), \theta=q^{\frac{1}{1- n}}$, $n\in 2\mathbb{N}+1$ and $q\in (0,1)$ are fixed. Define a function $h(t):=qt^{n}$. One can see that the $k$-th order iteration ...
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### Problem with a limit with a integral in it

Suppose that the temperature in a long thin rod placed along the $x$-axis is initially $\frac{C}{2a}$ if $|x| \leq a$ and $0$ if $|x| > a$. It can be shown that if the heat diffusivity of the rod ...
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### Does convergence of bounded, increasing sequences generalize from the set of real numbers to arbitrary partially ordered sets?

If $t$ is any real number, then I can find a strictly increasing sequence $(t_n)$ of real numbers converging to $t$. E.g. $t_n = t - \frac{1}{n}$ would do. Does this generalize to arbitrary partially ...
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### Finding the limit to infinity of a summation

The following problem was featured as a challenge in a previous exam paper and has left me stumped. Compute the following limit: $$\lim_{n \to \infty} \frac{1}{n^{2013}} \sum_{k=1}^n k^{2012}$$ ...
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### Determine whether the fourier series converges

I have calculated the Fourier Series of $g\left(x\right)=x$ on $\left(-\pi,\pi\right]$ extended periodically to $\mathbb{R}$ to be ...
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### Limit of indeteminate form $1^{(∞)}$

If we consider the function $f(x)=[(ax+1)/(bx+2)]^{x}$ where $a$,$b$ >$0$ and a I tried as follows]1 But at end i got stuck .
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### Finding the limits of a trig function

I have been struggling with finding the following limit: $$\lim_{x\to \pi} \frac{\cos x + 1}{x - \pi}$$ Use of L'Hospital's rule is not permitted. Thanks
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### Show that if $|s_{mn}-S|<\varepsilon$ then $|\lim_{n\to\infty}s_{mn}-S|\le\varepsilon$

This is the exercise 2.8.5 of the book Understanding analysis of Abbott. To put you in context I have that The sequence of partial sums $(s_{mn})$ is absolutely convergent, and the definition is ...
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### Show that sum is convergent [on hold]

Assume that $\sum\limits_{i=1}^{\infty} a_i = L$ . Show that $\lim_{x\to\infty} (S_1+S_2+...+S_x)/x = L$. Where $S_n=\sum\limits_{i=1}^{n} a_i$