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Calculus

Higher order derivativesCalculus

Introduction to infinite seriesCalculus

Functions expressed as power seriesCalculus

Taylor and maclaurin series- Home
- Integral Calculus
- Sequence and Series

Still Confused?

Try reviewing these fundamentals first.

Calculus

Higher order derivativesCalculus

Introduction to infinite seriesCalculus

Functions expressed as power seriesCalculus

Taylor and maclaurin seriesStill Confused?

Try reviewing these fundamentals first.

Calculus

Higher order derivativesCalculus

Introduction to infinite seriesCalculus

Functions expressed as power seriesCalculus

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Basic concepts: Higher order derivatives, Introduction to infinite series, Functions expressed as power series, Taylor and maclaurin series,

Related concepts: Linear approximation,

To approximate a function with a Taylor Polynomial of n’th degree centred around $a=0$, use

$f(x) \approx f(a) + f'(a)(x-a)+\frac{f^{"}(a)(x-a)^2}{2!}+ \cdots + \frac{f^n (a)(x-a)^2}{n!}$

where $P_n (x) = f(a)+f'(a)(x-a)+\frac{f^{"}(a)(x-a)^2}{2!}+ \cdots + \frac{f^n (a)(x-a)^2}{n!}$ is the Taylor Polynomial.

To find the difference between the actual value and your approximated value, look for the error term, which is defined as

$R_n(x)=\frac{f^{n+1}(z)(x-a)^{n+1}}{(n+1)!}$

Note that adding your Taylor Polynomial with your error would give you the exact value of the function. In other words,

$f(x)=P_n(x)+R_n(x)$

$f(x) \approx f(a) + f'(a)(x-a)+\frac{f^{"}(a)(x-a)^2}{2!}+ \cdots + \frac{f^n (a)(x-a)^2}{n!}$

where $P_n (x) = f(a)+f'(a)(x-a)+\frac{f^{"}(a)(x-a)^2}{2!}+ \cdots + \frac{f^n (a)(x-a)^2}{n!}$ is the Taylor Polynomial.

To find the difference between the actual value and your approximated value, look for the error term, which is defined as

$R_n(x)=\frac{f^{n+1}(z)(x-a)^{n+1}}{(n+1)!}$

Note that adding your Taylor Polynomial with your error would give you the exact value of the function. In other words,

$f(x)=P_n(x)+R_n(x)$

- 1.Approximating Functions with Taylor Polynomials and Error Bounds
i) Taylor Polynomials and the Error Term

- 2.Approximate ln 2 using the 3’rd degree Taylor Polynomial. Find the error term.
- 3.Find the 4th degree Taylor Polynomial centred around $a=0$ of $f(x)=e^x$. Then approximate $e^2$.
- 4.Find the 2nd degree Taylor Polynomial centred around $a=1$ of $f(x)=\sqrt{(x+1)}$ and the error term where $x \in [0,2]$.
- 5.Show that $f(x)=e^x$ can be represented as a Taylor series at $a=0$.
- 6.Show that $f(x)= \cos ?x$ can be represented as a Taylor series at $a=0$.

5.

Sequence and Series

5.1

Introduction to sequences

5.2

Monotonic and bounded sequences

5.3

Introduction to infinite series

5.4

Convergence and divergence of normal infinite series

5.5

Convergence & divergence of geometric series

5.6

Convergence & divergence of telescoping series

5.7

Divergence of harmonic series

5.8

P Series

5.9

Alternating series test

5.10

Divergence test

5.11

Comparison & limit comparison test

5.12

Integral test

5.13

Ratio test

5.14

Root test

5.15

Absolute & conditional convergence

5.16

Radius and interval of convergence with power series

5.17

Functions expressed as power series

5.18

Taylor series and Maclaurin series

5.19

Approximating functions with Taylor polynomials and error bounds