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The Integral Test

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The Integral Test

The Integral Test

The Integral Test

The Integral Test

Suppose we have a function f (x), defined for all x≥1, which is positive and decreasing. This function defines a sequence {f (n)} and a series

f (n) = f (1) + f (2) + ...    

Considering the following figure, we see that

f (2)≤f (x)dx    

since a rectangle with height f (1) and width 1 is contained within the region below the graph of f from 0 to 1.

Figure %: The Function Contains the Rectangular Regions

Similarly,

f (3)≤f (x)dx    

and so on. Thus we have

f (1) + f (2) + ... + f (n)≤f (1) + f (x)dx    

But the left side of this inequality is simply the nth partial sum for the series under consideration. If