Taylor's series

Question 1

Application of Taylor’s series

Answer 1

To get approximation

Question 2

Taylor’s series

Answer 2

Question 3

Linear approximation

Answer 3

Degree = 1
f(x) = f(a) + (x-a) f’(a)

Question 4

Second degree approximation

Answer 4

Question 5

In Taylor’s series, coefficient of (x-a)ⁿ

Answer 5

Question 6

The McLaurin series

Answer 6

The McLaurin series is a special case of the Taylor series where the expansion is done around x = 0

Question 7

Standard expansion 1

Answer 7

eˣ

Question 8

Standard expansion 2

Answer 8

sinx

Question 9

Standard expansion 3

Answer 9

cos x

Question 10

Standard expansion 4

Answer 10

log(1+x)

Question 11

Standard expansion 5

Answer 11

log(1-x)

Question 12

Standard expansion 6

Answer 12

log(1+x / 1-x)

Question 13

Power series

Answer 13

Question 14

Shortcut for nth-Degree Approximation of a Polynomial Function at x=0

Answer 14

Drop all terms with powers higher than n
Ex: 2nd degree approximation x³-3x²-5 at x=0 is -3x²-5

Question 15

How do you handle the indeterminate form 0/0 when expanding a function around a point using Taylor series?

Answer 15

To expand a function f(x) around a point where it leads to an indeterminate form like 0/0
* Substitute: Set x−c=y, where c is the point of expansion
* Rewrite the function in terms of and solve it

For ex:
f(x) = sin x / x-π at x= π
Here it is of the form 0/0 at x = π
let y = x-π
function will become f(y) = sin (π+y) / y