is there a convergent series which has a limit of 100
Is there a convergent series which has a limit of 100
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No sadly
[math] x(n)=\sum_{k=1}^n 9\cdot 10^{2-k} [/math]
Take your favourite convergent series and multiply every term by 100 divided by the original limit of that series.
[math]x(n)=100[/math]
wow holy shit BTFO
What if my favourite convergent series is
[eqn] \sum_{k=0}^\infty \left( \frac{k+1}{k^2 + 2k + 2} - \frac{k}{k^2 + 1} \right) [/eqn]
?
[math]\sum_{k=0}^\infty 100\left( \frac{\frac{k+1}{k^2 + 2k + 2} - \frac{k}{k^2 + 1}}{ \sum_{k=0}^\infty \left( \frac{k+1}{k^2 + 2k + 2} - \frac{k}{k^2 + 1} \right)}\right)[/math]
God damn it
[eqn] \frac{600}{\pi^2} \sum_{n=1}^\infty \frac{1}{n^2} [/eqn]
[math] 100\sum_k\left(\frac{0}{\sum_{k=1}^{\infty}0}\right) [/math]
If your favourite series is [math]\sum_{k=0}^\infty 0[/math], you deserve what you get.
>all these sub-70 IQ overcomplicating shit
[eqn]\sum_{n\,\geqslant\,0} 100\,\delta_{n,\,0}[/eqn]
gg ez
[math]\alpha\sum_{n=1}^\infty a_n[/math]
where (a_n) is a convergent series and alpha is a scalar that sets the limit to 100.
Not anymore. It commited suicide a month ago.
(1/n) + 100
or simply
100
>series
Yes 100 + 0 + 0...
A little less trivial? 99+.9 +.09 +.009...
What are constant series?
choose a sequence [math](a_n)[/math] such that [math]\sum_n a_n\to \ell\in\mathbf{R}\setminus\{0\}[/math].
then [math]\frac{100}{\ell}\sum_n a_n\to 100[/math].
divergent for all non-zero constants
100 followed by infinite zeros
I mixed up sequences and series.
[math]\displaystyle
-1200\times\sum_{n=1}^{\infty} n
[/math]
yes, [math]\sum_{k=1}^\infty \frac{(-1)^{k+1}}{k}[/math] under some rearrangement of terms
>is there a convergent series which has a limit of 100
The existence of a real number is equivalent to the existence of an infinite amount of series of rational numbers converging against it.
Yes.
p-series
anything ^n fraction if int is >1 it converges
Of course there is. There are many ways of proving it
the simplest is that you can make a series out of every sequence by summing differences of the terms…
the most fun way is with the Riemann theorem :
en.wikipedia.org
glorious
Let [math](a_n)_{n\in\mathbb{N}}[/math] be a real sequence defined by
[eqn]a_n = \begin{cases}100 &\text{if $n = 0$}\\ 0 & \text{otherwise}\end{cases}[/eqn]
Then the series
[eqn]\sum_{k=0}^\infty a_n[/eqn]
is convergent with limit 100.
ah yes, the series [math]x_1 = 100, x_n = 0 \forall n>1[/math]
how interesting...
how about infinite zeros followed by 100
My favourite convergent series is 0,0,0, ...
How to calculate factorial of real numbers (example 0.5 and sqr(2))? Explain pls or give me literature so I can do it myself
Sounds good to me
Gamma function