Limits Derivatives Integrals Infinite Series Parametrics Polar Coordinates Conics
Epsilon-Delta Definition
Finite Limits
One-Sided Limits
Infinite Limits
Trig Limits
Pinching Theorem
Indeterminate Forms
L'Hopitals Rule
Limits That Do Not Exist
Continuity & Discontinuities
Intermediate Value Theorem
Power Rule
Product Rule
Quotient Rule
Chain Rule
Trig and Inverse Trig
Implicit Differentiation
Exponentials & Logarithms
Logarithmic Differentiation
Hyperbolic Functions
Higher Order Derivatives
Slope, Tangent, Normal...
Linear Motion
Mean Value Theorem
1st Deriv, Critical Points
2nd Deriv, Inflection Points
Related Rates Basics
Related Rates Areas
Related Rates Distances
Related Rates Volumes
Definite Integrals
Integration by Substitution
Integration By Parts
Partial Fractions
Improper Integrals
Basic Trig Integration
Sine/Cosine Integration
Secant/Tangent Integration
Trig Integration Practice
Trig Substitution
Linear Motion
Area Under/Between Curves
Volume of Revolution
Arc Length
Surface Area
Moments, Center of Mass
Exponential Growth/Decay
Laplace Transforms
Describing Plane Regions
Infinite Series
Divergence (nth-Term) Test
Geometric Series
Alternating Series
Telescoping Series
Ratio Test
Limit Comparison Test
Direct Comparison Test
Integral Test
Root Test
Absolute Convergence
Conditional Convergence
Power Series
Taylor/Maclaurin Series
Radius of Convergence
Interval of Convergence
Remainder & Error Bounds
Fourier Series
Study Techniques
Choosing A Test
Infinite Series Table
Practice Problems
Exam Preparation
Exam List
Parametric Curves
Parametric Surfaces
Slope & Tangent Lines
Arc Length
Surface Area
Polar Coordinates
Slope & Tangent Lines
Arc Length
Surface Area
Conics in Polar Form
Vectors Vector Functions Partial Derivatives/Integrals Vector Fields Laplace Transforms Tools
Unit Vectors
Dot Product
Cross Product
Lines In 3-Space
Planes In 3-Space
Lines & Planes Applications
Angle Between Vectors
Direction Cosines/Angles
Vector Projections
Triple Scalar Product
Triple Vector Product
Vector Functions
Projectile Motion
Unit Tangent Vector
Principal Unit Normal Vector
Acceleration Vector
Arc Length
Arc Length Parameter
Vector Functions Equations
MVC Practice Exam A1
Partial Derivatives
Directional Derivatives
Lagrange Multipliers
Tangent Plane
MVC Practice Exam A2
Partial Integrals
Describing Plane Regions
Double Integrals-Rectangular
Double Integrals-Applications
Double Integrals-Polar
Triple Integrals-Rectangular
Triple Integrals-Cylindrical
Triple Integrals-Spherical
MVC Practice Exam A3
Vector Fields
Conservative Vector Fields
Potential Functions
Parametric Curves
Line Integrals
Green's Theorem
Parametric Surfaces
Surface Integrals
Stokes' Theorem
Divergence Theorem
MVC Practice Exam A4
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Prepare For Calculus 1
Ready For Calculus 2?
Trig Formulas
Describing Plane Regions
Parametric Curves
Linear Algebra Review
Word Problems
Mathematical Logic
Calculus Notation
Practice Exams
17calculus on YouTube
More Math Help
Tools and Resources
Academic Integrity
Learning/Study Techniques
Math/Science Learning
Memorize To Learn
Music and Learning
Instructor or Coach?
Math Books
How To Read Math Books

You CAN Ace Calculus

17calculus > infinite series > telescoping series

ATTENTION INSTRUCTORS: The new 2018 version of 17calculus will include changes to the practice problem numbering system. If you would like advance information to help you prepare for spring semester, send us an email at 2018info at 17calculus.com.

free ideas to save on books - bags - supplies

calculus motivation - music and learning

Join Amazon Student - FREE Two-Day Shipping for College Students

How to Read and Do Proofs: An Introduction to Mathematical Thought Processes

Telescoping Series

on this page: ► notation     ► solving

Telescoping Series are series whose partial sums have terms that cancel and may eventually converge. They are not very common in mathematics but are interesting to study to get a feel for partial sums and how series work.

The idea with telescoping series is to arrange the terms in a form where you can see what is canceling, then to take the limit of what is left. The best way to learn how to solve telescoping series problems is by example. First, let us look at a bit of notation. It will help you to follow the discussion if you have the infinite series table in front of you. We will refer to it several times.


Notice in the infinite series table, the series column for a telescoping series uses a little different notation than the other rows. We use \( b_n \) instead of \( a_n \) for the terms in the series. This is deliberate. What this is saying is that \( a_n = b_n - b_{n+1} \). An example of this is

\(\displaystyle{ \sum_{n=1}^{\infty}{\left( \frac{1}{n} - \frac{1}{n+1}\right)} = \left( 1 - \frac{1}{2}\right) + \left( \frac{1}{2} - \frac{1}{3}\right) + \left( \frac{1}{3} - \frac{1}{4}\right) + . . . }\)
where \( \displaystyle{ a_n = \left( \frac{1}{n} - \frac{1}{n+1}\right); ~~~ b_n = \frac{1}{n}; ~~~ b_{n+1} = \frac{1}{n+1} }\)

This is just one example of a telescoping series. They come in many forms. For example, you might see \( a_n = b_n - b_{n+3} \) in which case, every third term might cancel. When you find what you think might be a telescoping series, write out some terms until you see a pattern. The number of terms is determined by how far apart a term repeats. This is the main technique for handling telescoping series.

A key idea derived from this discussion is to watch the notation carefully, and when a textbook or teacher changes notation, this can mean that something is going on that might not be explicitly stated. Ask questions until you understand.


The main technique when determining convergence of telescoping series is to get an expression for the partial sum and taking the limit of the partial sum. This is shown in the following practice problem. Determine whether the series converges or diverges. If the series converges, find the value to which it converges, if possible.

Practice 1




You will probably not find a lot of discussion or videos on telescoping series since they do not show up very much. Your instructor may just touch on them in class but then may ask questions on the exam about them. Telescoping series are fairly straightforward but the main thing you need to watch for is the possibility of expansion by partial fractions. Most problems are not given as in the example above. You will probably have to do some algebra to get a fraction in telescoping form. After you do, the main technique is to write out several (usually many) terms until you see a pattern developing. It helps to build a table arranged to spot patterns. This will give you a feel for what is going on and enable you to answer questions about them. Check out the practice problems for examples and practice on working with these types of series.

A Note of Caution - In most series problems, you do not need to worry about where the series starts, i.e. it can start at \(0\), \(1\) or any other number. This is the case when you are trying to determine only if a series converges or diverges. However, when you are trying to determine what value a series converges to, you DO need to take into account what the starting value of the series is. So, keep track of this when you are working with telescoping series.

Search 17Calculus

Practice Problems

Instructions - - Unless otherwise instructed, determine whether the series converges or diverges. If the series converges, find the value to which it converges, if possible.

Basic Problems

Practice 2




Practice 3




Practice 4




Practice 5




Practice 6

\(\displaystyle{ \sum_{n=1}^{\infty}{\frac{4}{n(n+2)}} }\)


Practice 7

\(\displaystyle{ \sum_{n=1}^{\infty}{\frac{1}{n(n+1)}} }\)


Practice 8



Intermediate Problems

Practice 9




Practice 10

\(\displaystyle{\frac{5}{1\times 3}+\frac{5}{2\times 4}+\frac{5}{3\times 5}+ . . . }\)



Practice 11




Practice 12

\(\displaystyle{ \sum_{n=1}^{\infty}{\frac{1}{4n^2-1} } }\)



Real Time Web Analytics
menu top search practice problems
menu top search practice problems 17