Questions tagged [sequences-and-series]

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.

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36 views

How to solve this series？$\sum_{n=1}^{\infty}({\frac{1}{n^4}}\sum_{i=1}^{n}{\frac{1}{i}})$ [closed]

I only know the answer is $3\zeta(5)-\zeta(2)\zeta(3)$, but there is no detailed progress. $\sum_{n=1}^{\infty}({\frac{1}{n^4}}\sum_{i=1}^{n}{\frac{1}{i}})$
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Lower Semi Continuous Function Multiplied by a Bounded Positive Sequence.

Let $A:\mathcal{P}_2(\mathbb{R})\to \mathbb{R}$, be a lower semi-continuous functional on the space of Borel probability measures over the real line. Lower semi-continuous here means with respect to ...
24 views

A third term of a geometric progression, which has both positive and negative terms, is 1. The sum of its first three terms is 3. What's fourth term?

for clarity, I'll put the question down again due to lack of space The third term of a geometric progression, which has both positive and negative terms, is 1. The sum of its first three terms is 3. ...
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How to choose the $c$ term when evaluating the error of a Taylor series

I was looking at this link (https://personal.math.ubc.ca/~CLP/CLP1/clp_1_dc/ssec_taylor_error.html) which shows how to calculate the error of a Taylor series based on how many terms you wish to ...
55 views

Method for convergence with unknown constant

I was expanding on specific telescopic series $$\sum_{n=1}^\infty \frac{1}{n(n+1)}= 1$$ $$\sum_{n=1}^\infty \frac{1}{n(n+2)}= \frac{3}{4}$$ $$\sum_{n=1}^\infty \frac{1}{n(n+3)}= \frac{11}{18}$$ It ...
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Sequence $(x_n)$ of irrationals in $[0,1]$ such that sets $\{y + x_n: y \in \mathbb{Q} \cap [0,1]\}$ are disjoint.

Let $E = \mathbb{Q} \cap [0,1]$. Show there exists a sequence $(x_n)_{n \geq 1}$ in $[0,1]$ such that the sets $E + x_n = \{y + x_n: y \in E\}$ are disjoint. The sequence must consist of irrationals ...
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In an A.P , first term = 2 & sum of first 5 terms is 1/4th of sum of next five terms. Write the equation & find d.

Q: In an A.P , first term = 2 & sum of first 5 terms is 1/4th of sum of next five terms. Write the equation & find d. My solution: $S_5$ =$\frac{1}{4}$*{$S_{10}$- $S_5$}. Q says that it is ...
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Q regarding finding total sum at end of year with an increment till n-1th year.

Q: The income of a person is 3,00,000, in the first year and he receives an increase of 10,000 to his income per year for the next 19 years. Find the total mount, he received in 20 years. Currency = ...
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By using the digits 2, 3, 4 all positive integers are written in ascending form like 2, 3, 4, 22, 23, 24, .... Find the 124th number? [closed]

It is a mcq and the options are: A) 22342 B) 22222 C) 22223 D) 22232 I am practising for an olympiad.I am not understanding this...Can someone pls explain me this and give the solution if u can so
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Determine whether the series $\sum_{n=1}^\infty\left({\arctan(n)-\frac{\pi}{2}}\right)$ converge [duplicate]

I am trying to determine whether the series $$\sum_{n=1}^\infty \left({\arctan(n)-\frac{\pi}{2}}\right)$$ converges from my calculus homework. Im not sure which candidate to choose for comprasion ...