Σ (1/(3n+1)-1/(3n+2))
From 1 to ∞
#Summation
pine atlas
native escarpBOT
Σ (1/(3n+1)-1/(3n+2))
From 1 to ∞
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foggy ember
Σ (1/(3n+1)-1/(3n+2))
From 1 to ∞
Hm...
Well, you can show it converges, but I don't think it has a particularly simple value.
At least, I don't see a way of obtaining it.
pine atlas
Hm...
Well, you can show it converges, but I don't think it has a particularly s...
It's a question meant to be solved and obtain a specific value so there must be a way
foggy ember
I see. Hm...
brittle elm
Σ (1/(3n+1)-1/(3n+2))
From 1 to ∞
try to look at smaller “base cases” (i.e., sums of 1/(n+1)-1/(n+2) and 1/(2n+1)-1/(2n+2)) and observe what happens, and also what methods you use, to evaluate the sum in question
pine atlas
try to look at smaller “base cases” (i.e., sums of 1/(n+1)-1/(n+2) and 1/(2n+1)-...
Look as I put values on n it's not like an telescopic series but it kind of looks like expansion of ln(x) with terms multiple of 1/3 missing but the problem is that even terms have -ve on them so this idea failed
Then I thought of integration to get a approx value but I didn't reach the answer
brittle elm
Look as I put values on n it's not like an telescopic series but it kind of look...
what exactly did you integrate
pine atlas
what exactly did you integrate
The expression in sigma
foggy ember
Oh yeah, integration does work!
pine atlas
Oh yeah, integration does work!
Bruh what I think I did some error on my side
Can u do complete soln ??
I want to check for my mistake
foggy ember
Bruh what I think I did some error on my side
Can u do complete soln ??
I want ...
I mean, the integral is pretty easy to calculate.
brittle elm
The expression in sigma
yeah i see
cause you can approx with integral test
$\frac{1}{3n+1}=\sum_{k=0}^{\infty}(-1)^k(3n)^k$
versed bladeBOT
temu mod | promote to mod pls
brittle elm
foggy ember
$\frac{1}{3n+1}=\sum_{k=0}^{\infty}(-1)^k(3n)^k$
No, that doesn't work, since n can be big.
brittle elm
yeah
foggy ember
See my approach above. It's probably the intended one.
brittle elm
-1<n<1
sour cryptBOT
@pine atlas
feral heron
Σ (1/(3n+1)-1/(3n+2))
From 1 to ∞
$\sum_{n=1}^{\infty}\int_0^1\qty(x^{3n}-x^{3n-1})\mathrm{d}x=\int_0^1\frac{1-x}{1-x^3}\mathrm{d}x$
versed bladeBOT
wolfqz v1.3.2
feral heron
(the integral and sum are swappable in this case, by fubini's theorem)
pine atlas
is this solved?
Yes
And sorry for not closing
pine atlas
(the integral and sum are swappable in this case, by fubini's theorem)
The reason is bcz the intgrand converges right ??
feral heron
oh okay
feral heron
The reason is bcz the intgrand converges right ??
well there's more to it
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feral heron
**wolfqz v1.3.2**
it's absolutely convergent
sour cryptBOT
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feral heron
not ONLY convergent
pine atlas
it's absolutely convergent
Ohkkk