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The Binomial Series

Theory

For any real number kk and |x|<1, the binomial series is

(1+x)k=∑n=0∞(kn)xn=1+kx+k(k−1)2!x2+⋯ ,(1+x)^k = \sum_{n=0}^{\infty} \binom{k}{n} x^n = 1 + kx + \frac{k(k-1)}{2!}x^2 + \cdots,

where

(kn)=k(k−1)⋯(k−n+1)n!.\binom{k}{n}=\frac{k(k-1)\cdots(k-n+1)}{n!}.

Exercises

Exercise 1
Question

Find the power series of

f(x)=1(1+x)2f(x)=\frac{1}{(1+x)^2}

and its interval of convergence.

Show solution ↓
Solution

Step 1: Rewrite in binomial form.

1(1+x)2=(1+x)−2.\frac{1}{(1+x)^2} = (1+x)^{-2}.

Step 2: Apply the binomial series with k=−2k=-2.

(1+x)−2=∑n=0∞(−2n)xn.(1+x)^{-2} = \sum_{n=0}^{\infty} \binom{-2}{n} x^n.

Compute the coefficients:

(−2n)=(−1)n(n+1).\binom{-2}{n} = (-1)^n (n+1).

Step 3: Write the series explicitly.

1(1+x)2=∑n=0∞(−1)n(n+1)xn=1−2x+3x2−4x3+⋯\frac{1}{(1+x)^2} = \sum_{n=0}^{\infty} (-1)^n (n+1)x^n = 1 - 2x + 3x^2 - 4x^3 + \cdots

Step 4: Interval of convergence.

The binomial series converges when

|x|<1.(−1, 1)\boxed{(-1,\,1)}
Exercise 2
Question

Find the power series of

f(x)=14−xf(x)=\frac{1}{\sqrt{4-x}}

and its radius of convergence.

Show solution ↓
Solution

Step 1: Factor out the constant.

14−x=121−x4=12(1−x4)−1/2.\frac{1}{\sqrt{4-x}} = \frac{1}{2\sqrt{1-\frac{x}{4}}} = \frac12 (1-\tfrac{x}{4})^{-1/2}.

Step 2: Use the binomial series with k=−12k=-\frac12.

(1+u)^{-1/2} = \sum_{n=0}^{\infty} \binom{-1/2}{n} u^n, \quad |u|<1.

Here, let u=−x4u=-\frac{x}{4}.

Step 3: Substitute and write the series.

14−x=12∑n=0∞(−1/2n)(−x4)n.\frac{1}{\sqrt{4-x}} = \frac12 \sum_{n=0}^{\infty} \binom{-1/2}{n}\left(-\frac{x}{4}\right)^n.

Step 4: Radius of convergence.

Since

\left|-\frac{x}{4}\right|<1 \quad \Rightarrow \quad |x|<4,

the radius of convergence is

R=4.\boxed{R=4}.