How do you calculate the sum of these series | Mathematics | Page 1

peak totem Apr 14, 2024, 4:09 AM

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elfin etherBOT Apr 14, 2024, 4:09 AM

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peak totem

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thorn locust Apr 14, 2024, 4:42 AM

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Question 11 is directly asking you about the Leibniz Pi Approximation

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Question 15 is asking about the taylor expansion of e^x

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We can infer that x is 3, looking at the possible answers

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However, n = 2, so we know we should subtract something from e^3: the values of n for 0 and 1

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If n=0, 3^0 / 0! = 1

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If n=1, 3^1/1! = 3

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So it is e^3 - 1 - 3

peak totem Apr 14, 2024, 7:40 AM

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i dont think we learked about leibniz pi apprxomiation

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is it anything to do with integral remainder estimation or something like that

mild cradle Apr 14, 2024, 9:42 AM

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peak totem is it anything to do with integral remainder estimation or something like that

You're familiar with integration?

peak totem Apr 14, 2024, 4:06 PM

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yes

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this is cal 2 homework

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sorry was asleep

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we've also learned partial sums if it has anything to do with that? I must have missed this unit and cant find it

thorn locust Apr 14, 2024, 4:55 PM

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Have you learned taylor series?

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try the taylor expansion of pi*x, and relate that to q11

mild cradle Apr 14, 2024, 5:18 PM

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peak totem this is cal 2 homework

Okay, so. Are you familiar with the series sum(n = 0, inf) x^n and what it evaluates to?

peak totem Apr 14, 2024, 5:18 PM

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thorn locust Have you learned taylor series?

im covering this today so i think ill look into thos

peak totem Apr 14, 2024, 5:19 PM

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mild cradle Okay, so. Are you familiar with the series `sum(n = 0, inf) x^n` and what it eva...

no

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ik how to find convergence, radius, etc

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with series

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remainder too

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but didnt know how to calculate sum at all other than partial sums

mild cradle Apr 14, 2024, 5:19 PM

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Okay, so here's a trick.

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You know the difference of powers formula?

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@peak totem^

peak totem Apr 14, 2024, 5:22 PM

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yes

mild cradle Apr 14, 2024, 5:23 PM

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Okay, so if we have a finite sum of sequential powers of x, that is, sum(i = 0, n) x^i, how can the difference of powers formula help us evaluate that?

mild cradle Apr 14, 2024, 5:24 PM

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mild cradle Okay, so if we have a *finite* sum of sequential powers of `x`, that is, `sum(i ...

@peak totem (sorry if pinging you is annoying, you just don't respond if I don't so I don't know if you're paying attention otherwise.)

peak totem Apr 14, 2024, 5:25 PM

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no its totally okay i appreciate it so much

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Im not sure, but i really appreciate you pointing me in the right direction

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ill watch some videos on it

mild cradle Apr 14, 2024, 5:26 PM

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peak totem ill watch some videos on it

I mean, I can explain it all.

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I just need you to be engaged in telling me what you do and don't know or can or can't figure out so that I know what I need to explain and what you can do on your own.

peak totem Apr 14, 2024, 5:28 PM

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I feel like I just learn better from YT is the thing so I just wanted to know like what its called to solve these problems

mild cradle Apr 14, 2024, 5:33 PM

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peak totem I feel like I just learn better from YT is the thing so I just wanted to know li...

Okay, look. The short version is that (1 - x) * (sum (i = 0, n) x^i) = 1 - x^(n + 1), therefore sum(i = 0, n) x^i = (1 - x^(n + 1))/(1 - x). Take the limit as n -> inf and what do you get?

peak totem Apr 14, 2024, 5:35 PM

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infinity

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?

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having a hard time reading this notation

mild cradle Apr 14, 2024, 5:36 PM

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$\sum_{i = 0}^n x^i = \frac{1 - x^{n + 1}}{1 - x}$

crystal magnetBOT Apr 14, 2024, 5:36 PM

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Techie Literate

mild cradle Apr 14, 2024, 5:36 PM

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@peak totem This help?

peak totem Apr 14, 2024, 5:36 PM

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yes thanks

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1 sec

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infinity

mild cradle Apr 14, 2024, 5:41 PM

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Not always.

mild cradle Apr 14, 2024, 5:41 PM

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peak totem infinity

How did you determine that?

peak totem Apr 14, 2024, 5:42 PM

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hold on gtg need to take my mom to her appointment

mild cradle Apr 14, 2024, 5:42 PM

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peak totem hold on gtg need to take my mom to her appointment

Ping me when you're back.

peak totem Apr 14, 2024, 5:42 PM

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bet

peak totem Apr 14, 2024, 8:25 PM

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i found it in my notes im pretty syre

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mild cradle Apr 14, 2024, 8:48 PM

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peak totem

Right, that's exactly correct. And the cool thing about this is that we can use this and a bit of calculus to find series representations of functions that normally don't have them. For instance, ln(1 + x).

mild cradle Apr 15, 2024, 12:21 AM

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peak totem i found it in my notes im pretty syre

Actually, in hindsight it's seemingly just an easier way to derive the Taylor series. A Taylor series is a way to approximate any infinitely-differentiable function with a series: $f(x) = \sum_{n = 0}^\infty \frac{f^{(n)}(0)x^n}{n!}$

crystal magnetBOT Apr 15, 2024, 12:21 AM

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Techie Literate

mild cradle Apr 15, 2024, 12:22 AM

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Where $f^{(n)}(0)$ refers to the nth derivative of f evaluated at 0, and $f^{(0)}(x) = f(x)$.

crystal magnetBOT Apr 15, 2024, 12:22 AM

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Techie Literate

mild cradle Apr 15, 2024, 12:23 AM

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But honestly the sequential derivatives of some of these functions get annoyingly long, so I still prefer the shortcut derivation of the series.

peak totem Apr 15, 2024, 12:34 AM

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ohhh okay thank you so mcuh

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ill have to look into my notes more to see if thats something we cover

#How do you calculate the sum of these series