#question on series
brittle radishBOT
thorn cairn
no
you have the backwards
you have it backwards*
sorry
also they have to decrease exponentially to 0
i think the statement you are looking has the word monotonically instead of exponentially
vague lion
the Σ1/n² is convergent
but 1/n² doesn't exponentially decrease
but the reciprocal statement
if a sequence has an exponential decreasing , so its series converges
ocean cove
? 1/n^2 does exponentially decrease
thorn cairn
no it doesnt
it monotonically decreases
exponentially decreases would be something like (1/2)^n
ocean cove
exponentially decreases would be something like (1/2)^n
You literally just wrote the same equation
1/2^n = (1/2)^n
ocean cove
yes
vague lion
You literally just wrote the same equation
(1/2)^n is not the same as 1/n²
ocean cove
yes it is?
How
oh wait
Maybe in a series it’s different rather than a function
no im pretty sure there the same function
vague lion
no im pretty sure there the same function
in fact there are two different sequences
thorn cairn
write out the first few terms of 1/2^n and 1/n^2
you should see quickly they don't match up
ocean cove
1/2 , 1/4
1/2 , 1/4
vague lion
the first one 2 is at the base and n the exponent
and the second one : n is at the base and 2 at the exponent
the first one is way faster than the second
ocean cove
Am I washed or something 😭
vague lion
1/2 , 1/4
1/2 , 1/4
write more terms
use Desmos
ocean cove
ocean cove
write more terms
use Desmos
Look
vague lion
They exactly overlap
because you compare
(1/2)^x and 1/2^x which are the same by the rule of exponents
But here we study 1/x^2 and 1/2^x
thorn cairn
yep
if we go back to the original topic of this thread, you're on the right track about series convergence
im not sure if you've studied logic, but p => q does not necessarily mean q => p
ocean cove
?
vague lion
They exactly overlap
thorn cairn
?
your statement is backwards, so essentially if you have a convergent series with positive, decreasing terms, it does not necessarily imply that the sequence must be decreasing exponentially
ocean cove
Ya I see how that’s not equal
thorn cairn
however if the sequence is decreasing exponentially (and the terms are positive), then the series is convergent
ocean cove
your statement is backwards, so essentially if you have a convergent series with...
Oh okay
ocean cove
Ohh I see
okay let me reword it then
for a sum with positive decreasing terms the terms must decrease more than the last one to converge
Rightv
Right?*
thorn cairn
yep
you got it 👍
to reword it to make it more concise it could be
"for a sequence with positive, decreasing terms, each term must be less than the previous one for its infinite sum to converge"
hopefully this helped
ocean cove
to reword it to make it more concise it could be
"for a sequence with positive, ...
Well no not exactly
1/2 + 1/3 + 1/4 + 1/5 …
Is a sum that has terms being less that the previous one however it diverges to infinity
however it doesn’t decrease faster each time
thorn cairn
yeah oops sorry, thats my fault
ocean cove
im not sure if you've studied logic, but p => q does not necessarily mean q => p
I am familiar with this
ocean cove
yeah oops sorry, thats my fault
So the terms have to decrease more each time
thorn cairn
yes
if you really want the concise definition of what it means for a series to converge in relation to its sequence
ocean cove
I thought this meant exponential but I guess not
thorn cairn
the limit of the partial sums of a sequence must exist for its infinite series to converge
ocean cove
hm
thorn cairn
limit as n approaches infinity ofc
thorn cairn
if ur talking about a sequence that decreases exponentially strictly
ocean cove
right
thorn cairn
that is a sequence in the form of ar^n
if |r| < 1, then the series converges
where a is the first term of the series and r is the common ratio
ocean cove
if ur talking about a sequence that decreases exponentially strictly
Didn’t someone say it doesn’t have to be expotienl?
Although I thought x^2 is expotienal
ocean cove
x^2 is not exponential
hm
ocean cove
x^2 is not exponential
okay okay
thorn cairn
yeah so do you have a specific question you have orrrr
because there's a lot to talk about regarding series convergence that would probably be better suited for one of the channels
ocean cove
yeah so do you have a specific question you have orrrr
no im okay my original question was already awsnered
also how do you close this
oh there
odd rose
x^2 isn't exponential?
vague lion
x^2 isn't exponential?
this isn't an exponantial growth like e^x
because an exponential sequence is defined by the following rule : to get the next term you multiply by the same factor
x² is a polynomial growth.
which is slower
the factor to get the next term decereases each time
9/4 = 2,25 , 16/9 = 1,7777 , 25/16=1,5625 ...
rough quarry
.close