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…]
