Use Formula ( 17) for the binomial series to obtain the Maclaurin series for (a) (1)/(1+x)

Use Formula ( 17) for the binomial series to obtain the...

Key Concepts

Convergence of Power Series

The convergence of a power series, such as the binomial series, is determined by its radius of convergence. This concept defines the range of x values for which the series converges to the function it represents. Typically, for the binomial series expansion (1+x)^?, the series converges when |x| < 1, ensuring reliable approximation within this interval.

Generalized Binomial Coefficients

When the exponent ? is not a positive integer, the binomial coefficients are defined using a generalized formula that involves factorials or gamma functions. This generalization allows for the expansion of (1+x)^? into an infinite series, accommodating fractional or negative exponents. It is a key concept in the analysis and computation of series for a wide range of functions.

Maclaurin Series

A Maclaurin series is a Taylor series expansion of a function about zero. It expresses the function as an infinite sum of terms calculated from the derivatives of the function at zero. This series is extremely useful in approximating functions near x = 0 and is derived by setting up the Taylor series formula with a = 0.

Binomial Series

The binomial series is a power series expansion for expressions of the form (1+x)^?, where ? can be any real number, not just a positive integer. The expansion uses generalized binomial coefficients defined for arbitrary exponents, allowing the series to converge under certain conditions, typically for |x| < 1. This concept is central in obtaining series representations for a variety of functions.

Transcript

00:01 So for this particular problem, we are asked to use a formula for the binomal series to obtain the mclaren series for three different problems.

00:11 So for part a, we are doing this for 1 over 1 plus x, which we are going to rewrite as 1 plus x all to the negative first power.

00:24 And so what we want to do is we want to use the series.

00:29 So we're going to use 1 plus x to the m equals 1 plus m x plus m times m minus 1 x squared and then to the 2 factorial and then plus dot dot dot m times m minus 1 m minus k plus all over k factorial and this is the x to the k power.

01:08 So obviously we're not going to need to go quite that far.

01:12 But for this one, what i'm going to do is 1 plus x to the negative first power.

01:16 So this is going to be my m.

01:19 So it's going to equal 1 minus x plus, and then we have m times m minus 1 over 2 factorial, which is 2, x squared.

01:35 And i'm going to go ahead and go to the third one, and then i'm going to stop at that point for each one of these.

01:38 So then it's going to be m times m minus 1 times m minus 2 over 3 factorial x to the 3.

01:53 So then what i'm going to do is i'm going to simplify this down, so 1 minus x.

01:58 And then here i've got negative 1 times negative 2.

02:02 So that'll be 2 over 2 x squared.

02:07 And then here i've got negative 1 times negative 2 times negative 3.

02:17 So negative 1 times negative 2 is 2.

02:21 So it's going to be minus 6 over 3 factorial x to the 3rd.

02:29 And i simplify it down one more time.

02:30 So 1 minus x plus x squared.

02:33 And then of course 3 factorial is 6.

02:36 So it's going to be minus x to the 3rd.

02:38 So then to put this in a summation form, i'm alternating signs.

02:46 So negative 1 to the k, because the second term is and the fourth term.

02:52 So since we're starting at zero, it'll be fine.

02:56 And then we have x to the k power.

02:59 And this is the answer for that one.

03:02 So then for part b, we are looking at where f of x equals the cube root of 1 plus x...

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