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 infinite series power series taylor series

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

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17calculus > infinite series > remainder/error

Once you have found the infinite Taylor Series, you will often be asked to approximate the original function with a finite number of terms from the Taylor Series. Doing so introduces error since the finite Taylor Series does not exactly represent the original function. To handle this error we write the function like this.

$$\displaystyle{ f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + . . . + \frac{f^{(n)}(a)}{n!}(x-a)^n + R_n(x) }$$

where $$R_n(x)$$ is the remainder. Notice we are cutting off the series after the n-th derivative and $$R_n(x)$$ represents the rest of the series.

Lagrange's formula for this remainder term is

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

This looks very similar to the equation for the Taylor series terms . . . and it is, except for one important item. Notice that in the numerator, we evaluate the $$n+1$$ derivative at $$z$$ instead of $$a$$. So, what is the value of $$z$$? $$z$$ takes on a value between $$a$$ and $$x$$, but, and here's the key, we don't know exactly what that value is. So this remainder can never be calculated exactly. However, since we know that $$z$$ is between $$a$$ and $$x$$, we can determine an upper bound on the remainder and be confident that the remainder will never exceed this upper bound. So how do we do that?

Upper Bound on the Remainder (Error)

We usually consider the absolute value of the remainder term $$R_n$$ and call it the upper bound on the error, also called Taylor's Inequality.

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

How To Choose z

To find an upper bound on this error, we choose the value of $$z$$ using these rules.
1. If the $$n+1$$ derivative contains a sine or cosine term, we replace the sine or cosine term with one, since the maximum value of sine or cosine is one. This seems somewhat arbitrary but most calculus books do this even though this could give a much larger upper bound than could be calculated using the next rule. [ As usual, check with your instructor to see what they expect. ]
2. If we do not have a sine or cosine term, we calculate $$\abs{f^{(n+1)}(z)}$$ and then choose the value of $$z$$ between $$a$$ and the $$x$$-value that we are estimating that makes this term a maximum. Sometimes, we need to find the critical points and find the one that is a maximum. Since we have a closed interval, either $$[a,x]$$ or $$[x,a]$$, we also have to consider the end points. Many times, the maximum will occur at one of the end points, but not always.

Okay, so what is the point of calculating the error bound? The point is that once we have calculated an upper bound on the error, we know that at all points in the interval of convergence, the truncated Taylor series will always be within $$\abs{R_n(x)}$$ of the original function $$f(x)$$. This $$\abs{R_n(x)}$$ is a mathematical 'nearness' number that we can use to determine the number of terms we need to have for a Taylor series.

Here is a great video clip explaining the remainder and error bound on a Taylor series.

### Dr Chris Tisdell - What is a Taylor polynomial? [17min-25secs]

video by Dr Chris Tisdell

### Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on.

GOT IT. THANKS!

Instructions: For the questions related to finding an upper bound on the error, there are many (in fact, infinite) correct answers. However, for these problems, use the techniques above for choosing z, unless otherwise instructed. Give all answers in exact form, if possible.

Basic Problems

Find the fourth order Taylor polynomial of $$f(x)=e^x$$ at x=1 and write an expression for the remainder.

Problem Statement

Find the fourth order Taylor polynomial of $$f(x)=e^x$$ at x=1 and write an expression for the remainder.

Solution

### 383 solution video

video by PatrickJMT

Find the first order Taylor polynomial for $$f(x) = \sqrt{1+x^2}$$ about x=1 and write an expression for the remainder.

Problem Statement

Find the first order Taylor polynomial for $$f(x) = \sqrt{1+x^2}$$ about x=1 and write an expression for the remainder.

Solution

### 384 solution video

video by PatrickJMT

Intermediate Problems

Show that $$\displaystyle{ \cos(x) = \sum_{n=0}^{\infty}{ (-1)^n\frac{x^{2n}}{(2n)!} } }$$ holds for all x.

Problem Statement

Show that $$\displaystyle{ \cos(x) = \sum_{n=0}^{\infty}{ (-1)^n\frac{x^{2n}}{(2n)!} } }$$ holds for all x.

Solution

### 355 solution video

video by PatrickJMT

For $$\displaystyle{ f(x) = x^{2/3} }$$ and a=1; a) Find the third degree Taylor polynomial.; b) Use Taylors Inequality to estimate the accuracy of the approximation for $$0.8 \leq x \leq 1.2$$.

Problem Statement

For $$\displaystyle{ f(x) = x^{2/3} }$$ and a=1; a) Find the third degree Taylor polynomial.; b) Use Taylors Inequality to estimate the accuracy of the approximation for $$0.8 \leq x \leq 1.2$$.

Solution

### 356 solution video

video by PatrickJMT

Use the 2nd order Maclaurin polynomial of $$e^x$$ to estimate $$e^{0.3}$$ and find an upper bound on the error.

Problem Statement

Use the 2nd order Maclaurin polynomial of $$e^x$$ to estimate $$e^{0.3}$$ and find an upper bound on the error.

Solution

### 385 solution video

video by PatrickJMT

Determine an upper bound on the error for a 4th degree Maclaurin polynomial of $$f(x) = \cos(x)$$ at $$\cos(0.1)$$.

Problem Statement

Determine an upper bound on the error for a 4th degree Maclaurin polynomial of $$f(x) = \cos(x)$$ at $$\cos(0.1)$$.

Solution

### 386 solution video

video by MIP4U

Determine the error in estimating $$e^{0.5}$$ when using the 3rd degree Maclaurin polynomial.

Problem Statement

Determine the error in estimating $$e^{0.5}$$ when using the 3rd degree Maclaurin polynomial.

Solution

### 387 solution video

video by MIP4U

Estimate the remainder of this series using the first 10 terms $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{n^4+1}} } }$$

Problem Statement

Estimate the remainder of this series using the first 10 terms $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{n^4+1}} } }$$

Solution

### 1481 solution video

video by Krista King Math