# Tagged Questions

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### Relationship between $\int_a^b f(x) dx$ and $\sum_{i= \lceil a\rceil}^{\lfloor b\rfloor} f(i)$

Let we have a continuous function $f(x)$ in the interval $[ a,b ]$ Does there exist any relationship between its integral and summation of function-values defined at the integers between $a$ and ...
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### Showing that $\sum_{i=1}^n \frac{1}{|x-p_i|} \leq 8n \left( 1 + \frac{1}{3} + \frac{1}{5} + \cdots + \frac{1}{2n-1} \right)$

I'm taking a summer analysis course and preparing for our final exam later this week. Our professor gave us the following problem on our mock exam, and I can't seem to get anywhere on it. Does anyone ...
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### What does this infinite sum converge to?: $\sum_{n=1}^\infty \dfrac{1}{n^k} = \dfrac{1}{1^k} + \dfrac{1}{2^k} + \dfrac{1}{3^k} + …$

$$\sum_{n=1}^\infty \dfrac{1}{n^k} = \dfrac{1}{1^k} + \dfrac{1}{2^k} + \dfrac{1}{3^k} + \dfrac{1}{4^k} + \dfrac{1}{5^k} + ...$$ I've found that: when $k=1$, it diverge to infinity when $k=2$, it ...
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### Series $\sum \frac{1}{n^2\sin^3n}$

Question : Show that series $\sum \cfrac{1}{n^{2}\sin^{3}n}$ is divergent. Hint: Show that $$\sum \frac{1}{n|\sin(n)|}$$ is divergent. I am interested in other possible proofs for this question. ...
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### Lower bound of $\sum_{k = 1}^{N}1/(x + k)$

Let $f(x) := \sum_{k = 1}^{N}1/|x + k|$ for $x \in [0, N]$. Why is $f(x) \geq C\log N$ for all $x \in [0, N]$ where $C$ is an absolute constant. My work is: Since $x \in [0, N]$, we can remove the ...
Hey everyone this was give as a practice problem and i'm having trouble, any help is appreciated Let $(a_n)_{n=1}^\infty$ be a sequence such that $\displaystyle \lim_{n \rightarrow \infty} {a_n} = ... 2answers 66 views ### Is$f(x)=\sum_{k\in\mathbb N}\frac1k\sin\frac x{2^k}$bounded? $$f(x)=\sum_{k\in\mathbb N}\frac1k\sin\frac x{2^k}$$Is this function bounded? So obviously this converges because$|\frac1k\sin\frac x{2^k}|<|\frac x{2^k}|$and$\sum\frac x{2^k}$converges by ... 1answer 31 views ### Is$f$integrable in$L(X,\mathcal{X},\mu)$Is$f$integrable$L(X,\mathcal{X},\mu)\mu(E)=\sum_{n\in E\cap\mathbb{N}} |n^2+n-6|f:\mathbb{R}\rightarrow \mathbb{R_+}\cup\inftyf=(x-2)^{-4}$1answer 106 views ### Proof that$0 < \lim(a_n/b_n) < \infty$implies convergence/divergence of$a_n$and$b_n$Suppose that$a_n, b_n > 0$for all$n$. Prove that if$0 < \lim(a_n/b_n) < \infty$, then$\sum a_n$and$\sum b_n$either both converge or both diverge. I'm a bit unsure on how to proceed ... 1answer 52 views ###$\sum$over disjoint union of sets In Discrete mathematics, rule of sum says that "If a first task can be performed in$m$ways and another can be performed in$n$ways and two task be independent, then whole work can accomplished in ... 3answers 105 views ### Using Cauchy-Schwarz inequality to prove that the mean of n real numbers is less than or equal to the root-mean-square of those numbers Expressed mathematically, the question is to prove the that$\frac{1}{n}\sum_{i=1}^{i=n}{a_i}\leqslant\sqrt{\frac{1}{n}\sum_{i=1}^n{x_i}^2}.$First of all, what form of Cauchy-Schwarz should I ... 2answers 68 views ### Showing that$\sum \sqrt{a_na_{n + 1}}$converges given that$\sum a_n$converges [closed] Suppose a series$\sum a_n$of nonnegative reals converges; show that$\sum \sqrt{a_na_{n + 1}}$also converges. 0answers 42 views ### calculating sum of a limit of integral I am trying to calculate the following expression $$\sum_{m=0}^{\infty} \frac{1}{m!} \lim_{n \to \infty} \int_{\{(x,y):2x^2+y^2<n^2 \}}\left( 1 - \frac{2x^2+y^2}{n^2}\right)^{n^2} x^{2m}dx~dy ... 3answers 158 views ### How prove this limit \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{i+j}{i^2+j^2}=\frac{\pi}{2}+\ln{2} show that: this limit$$I=\lim_{n\to\infty}\dfrac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}\dfrac{i+j}{i^2+j^2}=\dfrac{\pi}{2}+\ln{2}$$My try: ... 4answers 218 views ### Determine if \sum_{n=1}^{\infty}\frac{(-1)^nn^2+n}{n^3+1} converges or diverges. Another series I found I'm struggling with. Determine if the following series converges or diverges.$$\sum_{n=1}^{\infty}\frac{(-1)^nn^2+n}{n^3+1}$$Ratio test and n-th root test are both ... 1answer 77 views ### The limit of a sum with Taylor's theorem I am trying the fallowing exercise : Let f\in C^1(\mathbb{R},\mathbb{R}) with f(0)=0 Compute the limite of : S_n=\sum _{k=0}^{n}f(\frac{k}{n^2}) I used Taylor's theorem , so I have$$ ... 3answers 640 views ### an inequality:$1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53n$is a positive integer, then $$1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53.$$ please don't refer to the famous$1+\frac1{2^2}+\frac1{3^2}+\dotsb=\frac{\pi^2}6$. I want to find a ... 0answers 24 views ### nonlinear sequence to sequence transformations i know matrix methods such as Cesaro,Holder,Riesz are regular linear sequence transformations. i wonder if there is any regular nonlinear sequence transformation? 2answers 73 views ### Help with compact notation for sum I've already understood the motive of this sum using the nom-compact way, but I want to do it in the compact way, so it will be rigorous. Please, I need some help: ... 1answer 41 views ### is this or (when) does this equality hold for weighted power series$s_n=\sum _{k=0}^n a_k$for every$n$and$x\in(0,1)$. Let$p=(p_n)$is a sequence of nonnegative numbers with$p_0>0$s.t$P_n= \sum_{k=0}^{n}p_k \rightarrow \infty$as$n \rightarrow \infty$... 2answers 54 views ### Determine whether$\sum_{n=1}^{\infty} \frac{x^2}{n^2}$converges uniformly on$[5,\infty)$I cannot figure out whether or not$\sum_{n=1}^{\infty} \frac{x^2}{n^2}$converges uniformly over$[5,\infty)$. My first thought was to try using the Weierstrass M-Test but failed immediately. Is ... 3answers 73 views ### Is a sequence convergent and if so what is the sum The sum $$\sum_{n=1}^{\infty} \frac{1}{(n+3)(n+2)}$$ Ive made it into partial fractions which gives$\frac{1}{n+2} - \frac{1}{n+3}$But im unsure how to tell if this now converges as obviously as ... 3answers 95 views ### Prove that$2\sum_{k=1}^n \cos(kθ) = \frac{\sin[\left(n+1/2\right)θ]}{\sin(θ/2)}-1\$ [closed]
Prove that $$2\sum_{k=1}^n \cos(kθ) = \frac{\sin[\left(n+1/2\right)θ]}{\sin(θ/2)}-1$$ By using $$e^{iθ}+e^{2iθ}+\cdots+e^{niθ}=\frac{e^{iθ}(1-e^{inθ})}{1-e^{iθ}}$$