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File Response: Points: Question Use 1 (1)/(\sqrt(8)) (1)/(\sqrt(27)) (1)/(\sqrt(64)) (1)/(\sqrt(125)) cdots to answer the following: a) (3 pts) Write the series identifying \sum_1^(\infty ) a_(n) b) (2 points) Write the first three partial sum (meaning S_(1),S_(2),S_(3) ) round to two decimal places if need it) c) (1 point) Name a theorem that could help identify if the series converges o diverge. d) (2 point) Use the theorem from part c to indicate if the series converges or not.

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Use 1 (1)/(\sqrt(8)) (1)/(\sqrt(27)) (1)/(\sqrt(64)) (1)/(\sqrt(125)) cdots to answer the following:
a) (3 pts) Write the series identifying \sum_1^(\infty ) a_(n)
b) (2 points) Write the first three partial sum (meaning S_(1),S_(2),S_(3) ) round to two decimal places if need it)
c) (1 point) Name a theorem that could help identify if the series converges o diverge.
d) (2 point) Use the theorem from part c to indicate if the series converges or not.

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file response points question use 1 1sqrt8 1sqrt27 1sqrt64 1sqrt125 cdots to answer the following a 3 pts write the series identifying sum1infty an b 2 points write the first three partial 00138

Added by Rachel A.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Sam Stansfield

09/02/2024

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File Response: Question Points: 8 Use $1 + \frac{1}{\sqrt{8}} + \frac{1}{\sqrt{27}} + \frac{1}{\sqrt{64}} + \frac{1}{\sqrt{125}} + \dots$ to answer the following: a) (3 pts) Write the series identifying $\sum_{1}^{\infty} a_n$ b) (2 points) Write the first three Show more…

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File Response: Points: Question Use 1 (1)/(\sqrt(8)) (1)/(\sqrt(27)) (1)/(\sqrt(64)) (1)/(\sqrt(125)) cdots to answer the following: a) (3 pts) Write the series identifying \sum_1^(\infty ) a_(n) b) (2 points) Write the first three partial sum (meaning S_(1),S_(2),S_(3) ) round to two decimal places if need it) c) (1 point) Name a theorem that could help identify if the series converges o diverge. d) (2 point) Use the theorem from part c to indicate if the series converges or not.

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Transcript

00:01 We want to find the limit of this sequence.

00:03 So here's the thing.

00:04 I'm going to define a as that limit.

00:12 Okay.

00:13 And i'm going to take the limit of both sides of this equation.

00:33 All right.

00:37 And so the left -hand side is just a.

00:42 And the right -hand side is this.

00:49 All right.

00:49 And then i'll square it.

00:57 All right.

00:57 And then i can subtract a -squared.

01:12 Both sides i get zero, which looks like it tells us a is minus three halves.

01:48 So here's the thing.

01:51 If we understand the square root to be the positive square root, this is never going to happen.

02:05 Because what's going to happen is every time that we take that square root, we're going to get a positive number.

02:11 We plug it back in there, we're going to get another positive number, which is actually bigger.

02:16 The only way to get to this is if we allow ourselves to take the negative square root.

02:27 But by definition, the square root is supposed to be positive.

02:35 Therefore, it diverges.

02:55 For the next one, so we're told that, all right, and we want to figure out this.

03:29 All right.

03:30 Well, i'm going to make life a little easier and rewrite this because x makes it confusing.

03:38 Because this x and that x are clearly not the same.

03:44 So i'm going to define this to be y like that.

03:59 Okay.

04:00 And then i'm going to define x equals 1 minus y.

04:08 All right...

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