Series

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How to Use the nth Term Test to Determine Whether a Series Converges

If the individual terms of a series (in other words, the terms of the series’ underlying sequence) do not converge to zero, then the series must diverge. This is the [more…]

How to Work with Geometric Series

Geometric series are relatively simple but important series that you can use as benchmarks when determining the convergence or divergence of more complicated series. A geometric series is a series of the [more…]

How to Analyze a p-Series

So-called p-series are relatively simple but important series that you can use as benchmarks when determining the convergence or divergence of more complicated series. A [more…]

How to Analyze a Telescoping Series

You don’t see many telescoping series, but the telescoping series rule is a good one to keep in your bag of tricks — you never know when it might come in handy. Consider the following series: [more…]

Using the Direct Comparison Test to Determine If a Series Converges

The direct comparison test is a simple, common-sense rule: If you’ve got a series that’s smaller than a convergent benchmark series, then your series must also converge. And if your series is larger than [more…]

How to Use the Limit Comparison Test to Determine Whether a Series Converges

The idea behind the limit comparison test is that if you take a known convergent series and multiply each of its terms by some number, then that new series also converges. And it doesn’t matter whether [more…]

Determining If a Series Converges Using the Integral Comparison Test

The integral comparison test involves comparing the series you’re investigating to its companion improper integral. If the integral converges, your series converges; and if the integral diverges, so does [more…]

Using the Ratio Test to Determine Whether a Series Converges

The ratio test looks at the ratio of a general term of a series to the immediately preceding term. The ratio test works by looking only at the nature of the series you’re trying to figure out [more…]

How to Use the Root Test to Determine Whether a Series Converges

The root test doesn’t compare a new series to a known benchmark series. It works by looking only at the nature of the series you’re trying to figure out. You use the root test to investigate the limit [more…]

How to Analyze Absolute and Conditional Convergence

Many divergent series of positive terms converge if you change the signs of their terms so they alternate between positive and negative. For example, you know that the harmonic series diverges: [more…]

How to Determine Whether an Alternating Series Converges or Diverges

An alternating series is a series where the terms alternate between positive and negative. You can say that an alternating series converges if two conditions are met: [more…]

Understanding Infinite Series in Calculus

In calculus, an infinite series is "simply" the adding up of all the terms in an infinite sequence. Despite the fact that you add up an infinite number of terms, some of these series total up to an ordinary [more…]

How to Find the Partial Sum of an Arithmetic Sequence

When your pre-calculus teacher asks you to calculate the kth partial sum of an arithmetic sequence, you need to add the first k terms. This may take a while, especially if [more…]

How to Find the Partial Sum of a Geometric Sequence

When your pre-calculus teacher asks you to find the partial sum of a geometric sequence, the sum will have an upper limit and a lower limit. The common ratio of partial sums of this type has no specific [more…]

How to Find the Value of an Infinite Sum in a Geometric Sequence

If your pre-calculus teacher asks you to find the value of an infinite sum in a geometric sequence, the process is actually quite simple — as long as you keep your fractions and decimals straight. If [more…]

How to Use Summation Notation to Show a Partial Sum of a Sequence

Summation notation is a useful way to represent the partial sum of a sequence. The sum of the first k terms of an arithmetic sequence is referred to as the [more…]

How to Expand a Binomial Whose Monomials Have No Coefficients or Exponents

The final outcomes of a binomial expansion depend on whether the original monomial had no coefficients or exponents (other than 1) of the variables. To find the expansion of binomials with the theorem [more…]

How to Find Binomial Coefficients

Depending on how many times you must multiply the same binomial — a value also known as an exponent — the binomial coefficients for that particular exponent are always the same. The binomial coefficients [more…]

How to Expand a Binomial Whose Monomials Have Coefficients or Are Raised to a Power

At times, monomials can have coefficients and/or be raised to a power before you begin the binomial expansion. In this case, you have to raise the entire monomial to the appropriate power in each step. [more…]

How to Expand a Binomial that Contains Complex Numbers

The most complicated type of binomial expansion involves the complex number i, because you're not only dealing with the binomial theorem but dealing with imaginary numbers as well. When raising complex [more…]

How to Test Whether a Series Converges or Diverges

Say you're trying to figure out whether a series converges or diverges, but it doesn't fit any of the tests you know. No worries. You find a benchmark series that you know converges or diverges and then [more…]

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