Mathematics Stack Exchange
Mathematics Stack Exchange

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.

Sequences and series are often considered as a main part of part of calculus, in addition to limits, continuity, differentiation and integration.

A sequence is an enumerated collection of objects in which repetitions are allowed. There are special types of sequences, such as arithmetic sequences (or arithmetic progressions), where the next term is a constant more than the previous; harmonic progressions, which is formed by taking the reciprocal of each term of an arithmetic progression; logarithmic progression, which is formed from a series whose progression getting smaller; and geometric progressions, where the next term is a constant multiplied by the previous term.

A sequence can be given by a direct formula (e.g. $a_n = 2^n + 3$), or by a recurrence relation. In a recurrence relation, the relation between the next term and the earlier terms is given. An example is the recurrence relation $F_{n+2}=F_{n+1}+F_n, n \geq 0.$ Together with the initial terms $F_0=0$ and $F_1=1$, this recurrence relation defines the famous Fibonacci sequence.

A series is formed by summing a sequence. A typical question is: When the number of summed terms goes to infinity, does the sum approach a finite limit? In other words, is it convergent? Several tests, such as the ratio test, the root test, the limit comparison test, the integral test, etc. can help to answer these questions.

Important note. Questions about guessing the next number in a sequence, with no explicit mathematical context, will usually be quickly closed. Consider posting these questions at Puzzling Stack Exchange instead.

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Determining if a series converges using the comparison test

I am using the comparison test to determine if series converge. I understand how to do it when there is $1$ in the numerator: $$\sum_{k=1}^\infty \frac{1}{\sqrt{4k^2-1}}$$ $$4k^2\gt 4k^2-1$$ $$\sqrt{4k^2}\gt \sqrt{4k^2-1}$$ $$2k\gt…
user70844
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$\sum_{n=1}^\infty (n!)^{-\frac{1}{n}}$

I have to study the character of this series $$\sum_{n=1}^\infty (n!)^{-\frac{1}{n}}$$ and I tried with the ratio test: $ \frac{a_{n+1}}{a_n}=…
Anne
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$a_{n}^{2} + a_{n} = 4S_{n} + 3 \ （n \geqslant 1）$，find $a_{n}$

Set sequence $a_{n} > 0(n \geqslant 1)$.$S_{n}$ is the sum of the preceding $n$ terms of the sequence $a_{n}$ and satisfies $a_{n}^{2} + a_{n} = 4S_{n} + 3$. 1)Find the general term formula of sequence $a_{n}$; 2)$b_{n} \overset{\text{def}}{=}…
Nothing of L_j_j
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$X(n)$ and $Y(n)$ divergent doesn't imply $X(n)+Y(n)$ divergent.

Please, give me an example where $X(n)$ and $Y(n)$ are both divergent series, but $(X(n) + Y(n))$ converges.
Walter r
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Does the series converge or diverge? (just need a way to approach the problem - no answer please)

$$\sum_{n=1}^{\infty}\frac{1}{n\sqrt{\ln(2n)}}$$ To I am trying to figure out exactly what to compare this series to in order to prove that it diverges. I know that $1/n$ diverges and I also know that by the p-series the square root diverges as…
EhBabay
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Taylor expansion for $f(x) = \sqrt{x}$ centered at $a=16$

What would be the simplest way to do this?
Billy Thompson
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Repetition in values of sequences of type $s_{n+1} = s_n + n^x / 10^n$

Assume we have a sequence of type $a_n=\frac{n^x}{10^n}$.The corresponding series would be one with the formula $s_{n+1}=s_n+\frac{n^x}{10^n}$. We can obviously see that when x = 1, this would result in $s_n$ approaching the value 0.123456... And…
Smiley1000
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Find $\left( \frac{2}{7} \right) \left(\log_2 \frac{y}{x} \right)$ if-

If $$x= (\cos1°) (\cos2°) (\cos3°) .............(\cos89°)$$ and $$y=(\cos 2°)(\cos 6°)(\cos 10°).............(\cos 86°)$$ Then what is the integer nearest to $$\left( \frac{2}{7} \right) \left( \log_2 \frac{y}{x} \right) ?$$ I tried putting it in…
user740036
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Infinite Geometric Series Sum

The sum to infinity of the series $$1+\frac2{11}+\frac3{121}+\frac4{1331}+\cdots$$ I tried finding the common ratio but I was not able to find it. I tried putting it in closed form but even that was not enough. Please help.
user740036
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A sequence defined by iterating over polynomials

$f(x)$ is a polynomial with integer coefficients.$a_{1}=f(0)$ ,$a_{2}=f(a_{1})$ and $a_k=f(a_{k-1}) \forall k \geq 3$, I have to show that if $a_{k}=0$ for some $k\geq 3$ then either $a_{1}=0$ or $a_{2}=0$. My attempt: Say for example we have…
Soumyadip Sarkar
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Series whose terms are implicitely defined

Consider the next statement: Let $(u_n)_{n \in \mathbb{N}^*}$ be defined implicitely by: $$u_n = \tan(u_n) \\ -\frac{\pi}{2}+n\pi\leq u_n\leq\frac{\pi}{2}+n\pi$$ Then: $$\sum_{n=1}^{\infty} \frac{1}{u_n^2}=\frac{1}{10}$$ I found this statement in…
KRPO
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Determine if the Series Converges and Explain approach

Is the series Convergent? Explain and find Sum:- $$\sum_{n=1}^{\infty} \ (2)^\frac{1}{n}-(2)^\frac{1}{n+1}$$ Thanks
Walker
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Evaluate: $\sum_{n=2}^\infty \frac{n(n-1)}{2^{n-2}}$

How would I go about finding the sum of this series? $$\sum_{n=2}^\infty \frac{n(n-1)}{2^{n-2}}$$
Billy Thompson
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Comparison test for trig series

I have following series $$\sum_{n=1}^\infty \frac{\sin(3n)}{n^4} $$ Then I use comparison test, compare them to $$\sum_{n=1}^\infty \frac{1}{n^4} $$ And conclude they converge. However, I have this task marked as mistake "You can do comparison test…
Arnie
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Show that the series diverges

I don’t know how to prove that : Let $(a_n) \in ( \mathbb R^{+*})^\mathbb{N}$ We assume that $\forall n \in \mathbb N ,a_{n+1}
abc
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