Binomial Theorem

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Introduction

Notice

Using Sigma Notation

Example

The Special Case ( 1 + x ) n

Example (1 + x ) 6

What about when n is not a natural number?

The Binomial Series

Let f(x) = ( 1 + x ) n, n ЄR-(NU{0}) Let’s find the Maclaurin series for f f (k) (0)f (k) (x)k 1( 1 + x ) n 0 nn( 1 + x ) n-1 1 n(n-1)n(n-1)( 1 + x ) n-2 2 n(n-1)(n-2)n(n-1)(n-2)( 1 + x ) n-3 3 n(n-1)(n-2)(n-3)n(n-1)(n-2)(n-3)( 1 + x ) n-4 4 n(n-1)(n-2)(n-3)….[n-(k-1)] n(n-1)(n-2)…..….[n-(k-1)] ( 1 + x ) n-k k

Thus the Maclaurin series for (1+x) n

Thus the Maclaurin series for f

Finding the interval of convergence of this series:

Example Expand and then use the expansion to obtain a rough estimation of 1/√2

Solution

Approximating 1/√2

Homework