# Questions tagged [sequence-of-function]

Use this tag only when your query is about sequences of functions. Don't use this tag for any other sequence such as sequences of real numbers or sequences of complex numbers etc.

240 questions
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### Uniformly Bounded and Bounded Variation [on hold]

Studying functions of bounded variation, the following exercise came up: Let $(f_n)_{n\in \mathbb{N}}$ be a sequence of functions with $f_n:I \to \mathbb{R}$. Show that: If $(f_n)_{n\in \mathbb{N}}$...
3answers
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### Limit of a sequence as $n\to\infty$ [closed]

Let $x_{0}$ be a positive real number and $n\in\mathbb{N}$. Then what is $$\lim_{n\to\infty}\{(x_0+n)^r-n^r\}$$ where $r\in (0,1)$ is fixed number.
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### Calculate $\lim_{n \to \infty} \int_0^1nx^nf(x)dx$. [duplicate]

Consider the function $f$ which is continuous. Calculate $\lim_{n \to \infty} \int_0^1nx^nf(x)dx$. Here first I attempted to prove $f_n=nx^n$ is uniformly convergent using sup-norm limit but ...
0answers
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### Sum of an odd recursive sequence

Let $a_0 = 1$ $a_1 = 1 - \frac{e}{2}$ $a_n = \frac{e}{2^n} - \frac{1 - a_{n-1}}{n - 1}$ for $n > 1$. Find $\sum_{r=0}^{\infty}a_r$.
1answer
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### Is it a continuous?

For each $n\in\mathbb{N}$, define a function $f_n:[0,1]\to\mathbb{R}$ by $$f_n(x)=\int_{1/n}^{1}\frac{t^{x}}{\sqrt{t+x}}\,dt.$$ Then, is it continuous for each $n\in\mathbb{N}$? I don’t understand ...
2answers
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### Uniform convergence of a sequence of functions 4

Prove that the sequence $\left((nx)/(1+4n^2x^2)\right)_{n\in\mathbb N}$ is not uniformly convergent on $(-a,a)$, where $a > 0$ My attempt: $\lim_{n\to\infty}(nx)/(1+4n^2x^2) = 0 = f(x)$ Now, ...
1answer
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### Pointwise convergence of $h_{n}(x)$ on [0,$\infty$)

I know that it converges pointwise to $1$ if $x>0$ and to $0$ if $x=0$ using limits . But I am struggling to show this formally. Any help would be greatly appreciated . Thanks
1answer
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### How to show a function is locally C^1 implies globally C1?

Actually there is a series problem like f(x)=sum(n=1 to ∞)[sin(nx^2)/1+n^3], the question was whether f(x) is C^1 or not. This question has already answered, but a big issue of mine is I can't find ...
1answer
38 views