Find a power series representation for f, and graph f and several partial sums sn(x) on the same

step 1 So here is the power series for the tangent inverse of x function. So tangent inverse of x is eq

Find a power series representation for f, and graph f and...

Key Concepts

Graphical Analysis of Series Approximation

Graphing both the function and its partial sums provides a visual demonstration of convergence. As the number of terms in the partial sums increases, the graphs of these sums appear to more closely follow the graph of the original function, revealing the effectiveness of series approximations.

Partial Sums and Convergence

Partial sums are the sums of a finite number of terms from a power series that approximate the function. As more terms are added, these sums tend to converge towards the actual function, illustrating how the series progressively refines the approximation.

Power Series Representation

A power series expresses a function as an infinite sum of terms involving powers of the variable multiplied by coefficients. This representation enables the approximation of complicated functions within a specific interval, making it a powerful analytical tool in calculus.

Taylor Series

The Taylor series is the method of representing a function as an infinite sum of terms calculated from its derivatives at a specific point. This concept forms the foundation for approximating more complex functions using polynomials, which become increasingly accurate as more terms are included.

Interval of Convergence

The interval of convergence is the range of input values over which the power series converges to the function it represents. It ensures that the series approximation is valid and reliable by specifying where the sum of the series accurately reflects the original function.

Transcript

00:04 So here is the power series for the tangent inverse of x function.

00:10 So tangent inverse of x is equal to x minus x cubed over 3 plus x to the 5th over 5, minus x to the 7th over 7, plus an infinite more number of terms.

00:24 And this is the power series representation for tangent inverse of x as long as x is in the interval from negative 1 to 1.

00:35 We want to find the power series representation for the inverse tangent of 2x.

00:50 And this is easily done.

00:52 If tangent inverse of x equals this series, then tangent inverse of 2x.

01:00 To find a series for this function, simply substitute in 2x everywhere you see x.

01:07 So tangent inverse of 2x will be 2x minus and then substitute 2x in for the x.

01:20 So 2x to the 3rd over 3 plus 2x to the 5th over 5 minus.

01:43 Notice the plus and subtraction signs keep alternating.

01:47 So tangent inverse of 2x minus 2x to the 7th over 7 plus, and of course the series goes on indefinitely.

02:07 So now we just got to simplify 2x to the 3rd, 2x to the 5th, 2x to the 7th.

02:15 So we have 2x minus, well 2x to the 3rd is really 2 to the 3rd times x to the 3rd, 2 to 3rd is 8.

02:26 So we have 8 over 3 times x cubed plus.

02:35 And then 2x to the 5th.

02:37 2 to the 5th is 32.

02:39 So we'll have 32 x to the 5 over 5.

02:44 32 x to the 5 over 5.

02:49 And we can just put the 32 over to 5 minus 2 to the 7th.

02:56 Double check on the calculator.

02:58 2 to the 7th power.

02:59 Is 128.

03:02 So we're going to have 128 times x to the seventh over seven plus obviously an infinite more number of terms.

03:16 So this is the power series representation for the tangent inverse of 2x.

03:23 Now we can take partial sums of this series to approximate the tangent inverse of 2x.

03:33 Function.

03:34 So a sum with just one term would be 2x.

03:37 The sum with two terms would be 2x minus 8 thirds x cubed.

03:42 If you wanted to use a sum with three terms to approximate the function, then you just use the first three terms in the series.

03:51 And likewise, if you want to use a sum with four terms to approximate the tangent inverse of 2x function, use all four of these terms.

04:00 So what we're going to do next is we're going to graph the tangent inverse of 2x function on the graphing calculator using desmos.

04:08 Then we're going to graph some partial sums.

04:12 So let's use a sum.

04:17 We'll graph the tangent inverse of 2x function, and then we'll graph the partial sums.

04:29 So s2 of x simply means the partial sum of this power series using the first two terms.

04:39 So we will graph 2x minus 8 thirds x cubed...

Please give Ace some feedback