Within the interval of convergence of the seties, let f(x)=1-x+x^2-x^3+… Estimate the valu

Within the interval of convergence of the seties, let ...

Key Concepts

Geometric Series

A geometric series is a sequence of terms where each term is produced by multiplying the previous term by a constant called the common ratio. Its general form is a + ar + ar^2 + ar^3 + …, and understanding its structure is fundamental in analyzing series behaviors.

Interval of Convergence

The interval of convergence refers to the set of input values for which an infinite series converges (i.e., approaches a finite limit). For a geometric series, convergence is assured when the absolute value of the common ratio is less than one.

Infinite Geometric Series Sum Formula

When the common ratio of a geometric series satisfies |r| < 1, the series converges to a sum given by the formula a/(1 - r), where a is the first term. This formula provides a powerful tool for evaluating the series efficiently.

Transcript

00:01 In this question we are given the function fx equal to 3 divided by x squared minus x minus 2.

00:08 And we know that the denominator we can write it out as a factor of the form x minus 2 times x minus 1 plus 1 here.

00:20 And now here we can use the partial fraction to rewrite this fx as the sum constant a over the x minus 2.

00:30 Plus some constant b over x plus 1.

00:35 So doesn't imply that the a times x plus 1 plus the b times x minus 2 will must equal to this value 3 here.

00:47 And now if we fix the x equal to minus 1, it means that the first term here equal to 0, and then we will have the second terms becomes the minus 3b.

01:00 Equal to 3 therefore p here equal to minus 1 and now the second case if x equals to 2 and then the the first term here will become the 3a and the second term here equal to 0 so we have the 3a equal to 3 therefore the a here equal to 1 as a result we get the fx equal to 2 1 over x minus 2 minus 1 over x plus 1.

01:31 Here we can rewrite this down as the for the first fraction i will divide everything by the minus 2 therefore we have a minus 1 over 2 divided by 1 minus x over 2 and now for the second case i will have the 1 over 1 i will write the time as a minus minus x so we will recall that we have the 1 over 1 minus x and will equal to the same summation of the x power n from zoh to infinity and it's valid for the absolute x smaller than 1.

02:08 Therefore, using this formula here, we can get the fx, it will equal 2.

02:15 We keep the minus 1 out of 2 outside, and in here we will replay this x here by this x over 2.

02:24 So we have the summation of the x over 2, bow of the n, and now gets from 0 to infinity.

02:31 Now for the second case here, we will replace this x by the minus x here.

02:37 Therefore we have the summation of the minus x power of the n...

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