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.
65,956
questions
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How do I solve this Sum? [closed]
Calculate the following sum:
$$\sum_{n=1}^{\infty} \arctan \left( \frac{x}{1+n(n-1) x^2} \right)$$
Using that:
$$\arctan a-\arctan b=\arctan \left( \frac{a-b}{1+a b} \right)$$
- 11
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votes
3
answers
108
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Is there exact formula or recursive relation for the sequence $a_{n+1}=a_n + n $? [closed]
Good evening to everybody.
I was reading a chapter today on a book related to recursive sequences, and I saw as an example the following simple sequence : $$a_{n+1}=a_n + n $$
I was thinking if we ...
- 374
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1
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views
Finding minimum of geometric sequence
For the given natural number $0<n\in\mathbb{N}$, define the function as follows:
$$f\left(r\right)=2\cdot\sum_{i=0}^{n+2}r^{i-n}+1$$
I want to find the minimum for that function in the domain $r\in\...
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1
answer
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Solutions of $((a-1)!)^x=0$
I have an equation that I couldn't solve. I will be glad if you help me to do it.
Are there solutions in:
$((a-1)!)^x=0,$ with $a\in\mathbb N?$
Because with my knowledge in math, I found no solutions.
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votes
4
answers
885
views
How do I write the rule for this sequence that is neither arithmetic nor geometric as an equation? [closed]
I have the sequence $3, 5, 9, 17, 33, 65,\dots$
I know the rule is multiply by $2$ then subtract one. $3$ times $2$ is $6$ minus $1$ is $5$ and so on. Easy enough.
How do I write the rule for this ...
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votes
1
answer
97
views
Showing that $i^n=0$ when $i$ is the imaginary unit and $n$ is infinite [closed]
Let be the sum:
$$\begin{align}
s&= i+i^2+i^3+\cdots+i^n\\
&= i-1-i+1+i-\cdots\\
&=(1+i)(1-1+1-1+\cdots)
\end{align}$$
As Grandi's sum is equal to $1/2$, so: $s=\frac12(1+i)$.
But, $s$ ...
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1
answer
40
views
Evaluating an infinite series. [closed]
Find $$1 + 2 + 4 + 8 + \cdots$$
Now I let $$s = 1 + 2 + 4 + 8 + \cdots$$
So,
$$s = 1 + 2(1 + 2 + 4 + \cdots)$$
$$s = 1 + 2s$$
and this gives us:
$$s = -1$$
Is there a more rigorous proof for this ...
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3
answers
90
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Calculating the limit of the sequence $a_n=(2n+3)^\frac{1}{n}$ without using calculus.
Calculating the limit of the sequence $a_n=(2n+3)^\frac{1}{n}$ without using calculus. I know that $a_n\to 1$, but I found it using calculus. However, this is an example for a real analysis course and ...
user506873
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1
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infinite series containing cos(x/n)
for $n \in \{ 1,2,3,\dots \}$ we have $\sum\frac{(-1)^{n+1}\cos(\frac{x}{n})}{n}$. If integrated term by term twice, the resulting series clearly diverges. Does this mean the original series ...
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1
answer
53
views
First day income is 1 dollars . On second day income is 4 dollars. And on third day income is 10 dollars. What is the total income after 30 days? [closed]
First day income is $1$ dollars . On second day income is $4$ dollars. And on third day income is $10$ dollars. What is the total income after $30$ days ?
The answer is given $1335$. But how can I ...
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1
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Could be $(x_n)_n $ convergent? [closed]
Let $(x_n)_n $ such that $\displaystyle\frac {2018}{x_n} =\left(\frac {x_{n+1}}{x_n}\right)^{(n+1)/2}$, $\forall n\geq 1$.
I have to study the convergence of the sequence.
I have no idea how to ...
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1
answer
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views
find the value of the sumation? [closed]
find the value of the limit
$$\lim_{n\to\infty}\frac{2^{-n^2}}{ \sum_{k=n+1}^{\infty} \frac{1}{2^{k^2}}}=\infty$$
a) $0$
b) $some c ∈ (0,1)$
c) 1
d) ∞
this is the orginial questions
from my ...
- 14.6k
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2
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How to evaluate this sum? $ \sum_{k=1}^n \frac{12^k}{(4^k - 3^k)(4^{k+1} - 3^{k+1})} $
How to solve this exercise?
Evaluate
$$
\sum_{k=1}^n \frac{12^k}{(4^k - 3^k)(4^{k+1} - 3^{k+1})}
$$
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votes
1
answer
61
views
Sum of a series ... [closed]
I have a series as- (2)(2) + (4)(4)+ (7)(8)+ (11)(16)+(16)(32)+... and I can't successfully figure out its sum.I tried finding the nth term and arranging it into a difference.Any help would be highly ...
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2
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At what x does this series converge [closed]
At what $x$ does the series $\sum_{0}^{\infty} (1-(\frac{\sqrt{x}}{3}))^n $ and prove for those $x$ it converges to $f(x)=\frac{3}{2+ \sqrt{x}}$ .
I know the general term must tend to zero in order ...
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