Convergence of Series: Integral Test
If we start with a decreasing and positive function
You can define And then you get a series of positive terms
In general, you can say that this inequality holds: In the mathlet you can play with a particular case Dragging the dots you can change the values of lambda and p.
In theses cases, the series that you get is like a pseries (translated and expanded). Some of these integrals diverge then the series diverges too: This is the case when in a pseries p es equal or greater than 1. The integral and the series diverge if you 'crosses this line' dragging the green dots: In some other cases, the integral and the series converges: In the mathlet, click the play button to see the animation. You can see that the series is the integral plus something that is less than a_{k}. And we can say that Better than that, you get a lower and an upper bounds of the series. For example, consider the integral Then The series converges and the lower and upper bounds are
REFERENCES
These are classical results but this page is directly inspired in Jim Fowlerīs lesson 'How can integrating help us to address convergence'.
This is part of the Coursera course Calculus Two: Sequences and Series that Jim Fowler is teaching
with all his enthusiasm (October 2013).
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