Maclaurin Series Formula

Maclaurin series formula

In maths, we come across several complicated functions that are difficult to solve. But we have some standard functions that help us approximate the more complicated functions. 

One such approximate function is Taylor’s series. Taylor’s series gives us the value of a function at a point (a). Maclaurin series is a special case of Taylor’s series that gives us the value of a function at a point (0).

The formula for the Maclaurin series

The Taylor’s series is given by the formula

f(x) = f (x) + f’ (x) * x + f’’ (x) * x²/ 2! + f’’’ (x) * x³/ 3! + …

now putting f (x) = f (0) in the Taylor’s series we get the Maclaurin series

f(0) = f (0) + f’ (0) * x + f’’ (0) * x²/ 2! + f’’’ (0) * x³/ 3! + …

Ex.1. Find the Maclaurin series of ex. 

Here f (x) = e^x

f (0) = e⁰ = 1

f’ (x) = e^x

f’ (0) = e⁰ = 1

f’’ (x) = e^x

f’’ (0) = e⁰ = 1

by the Maclaurin series formula we have, 

f(0) = f (0) + f’ (0) * x + f’’ (0) * x²/ 2! + f’’’ (0) * x³/ 3! + …

f(0) = 1 + 1 * x + 1 * x² / 2 + 1 * x³ / 6 + … 

f(0) = 1 + x + x² / 2 + x³ / 6 + …

Ex.2. Find the Maclaurin series of sin x.

Let f (x) = sin x

f (0) = sin 0 = 0

f’ (x) = cos x

f’ (0) = cos 0 = 1

f’’ (x) = – sin x

f’’ (0) = – sin 0 = 0

f’’’ (x) = – cos x

f’’’ (0) = – cos 0 = – 1

by the Maclaurin series formula we have, 

f(0) = f (0) + f’ (0) * x + f’’ (0) * x²/ 2! + f’’’ (0) * x³/ 3! + …

f(0) = 0 + 1 * x + 0 * x²/ 2! – 1 * x³/ 3! + …

f(0) = x – x³/ 3! + x⁵ / 5 + …