Determine the sums of the following geometric series when they are convergent. (1)/(5)+(1)/(5^4)

Determine the sums of the following geometric series when...

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Key Concepts

Geometric Series

A geometric series is a series where each term is produced by multiplying the previous term by a constant factor known as the common ratio. This structure allows the series to be analyzed using specific formulas that relate the first term and the common ratio to the sum of the series, especially when dealing with infinite sums.

Convergence of Infinite Geometric Series

For an infinite geometric series to converge, the absolute value of the common ratio must be less than one. This condition ensures that the successive terms become progressively smaller and approach zero, allowing the infinite sum to approach a finite limit.

Sum Formula for Infinite Geometric Series

When an infinite geometric series converges, its sum can be calculated using the formula S = a/(1 - r), where 'a' represents the first term and 'r' the common ratio. This formula provides a direct way to find the total sum of all terms in the series as long as the convergence criterion is met.

Transcript

00:01 One of the first things that we might notice for this one is that the powers each increase by three.

00:08 So one over five to the one.

00:11 And then from one to four plus three, four to seven, plus three, seven to ten plus three.

00:18 Okay, so we notice something going on there, some pattern.

00:21 So then as we start to write this geometric series, usually one of the first things is to start off with whatever our first term is.

00:28 If it's something other than one, that's usually a coefficient in front.

00:35 So let's leave that one -fifth out front.

00:38 Then let's see what we got.

00:39 Well, we got a one -fifth here.

00:42 But to generate the second, well, let's just start with the first term, right? well, it's just going to be to the power zero here.

00:49 And then that would get us the first term.

00:52 So no problem.

00:53 But what if it's now 1 over 5 to the 4th? well, we need a total of four powers.

00:58 So what if we do that this one has three powers of n and then combining it back with the first one that would give us one over five to the four if n is one.

01:12 So we're going to start our index at zero and it's going to go off to infinity.

01:18 Now let's just go through a thought process here and say, okay, hopefully this works.

01:23 I'm just going to move this over a little bit...

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