It is not always easy to find the sum of an infinite series. But with the help of the Integral Test, we can not only determine whether a series is convergent or divergent but also find suitable estimates of the sum.
The Integral Test takes an infinite series and transforms it into an Improper Integral. In doing so, we can approach the infinite series like we would a problem where we are asked to find the area under the curve. And therefore, we can evaluate the improper integral as a limit of the partial sums.
But there are a few requirements to using the Integral Test.
We must first determine that the series is a continuous, positive and decreasing function. And secondly, it needs to be “easy” to integrate!
If so, then we can determine convergence or divergence by using Improper Integrals.
Now here are a few helpful hints for when we change our series into an improper integral.
- The lower limit on the improper integral must be the same value that starts the series.
- The function does not actually need to be decreasing and positive everywhere in the interval.
As Paul’s Online Notes accurately states, all that is required is for the function to eventually be decreasing and positive.
And here’s an interesting fact about the Integral Test… it will help us to corroborate what we know about a p-series!
Moreover, it will be helpful for us to determine convergence whenever our series includes “ln(n)” or any other positive, decreasing function that is easy to integrate.
Integral Test Video
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