How does the integral test work (partial sums)?

Question

anonymous · Answer

The Integral Test states that when a_n = f(n):
(i) If the integral from 1 to infinity of f(x) is convergent, then the partial sum of a_n is convergent.

anonymous · Answer

(ii) If the integral from 1 to infinity of f(x) is divergent then the partial sum of a_n is divergent.

anonymous · Answer

Could someone explain to me how this works or rather why it works?

amistre64 · Answer

what is the continuous version of a discrete sumation?

amistre64 · Answer

an hour ago eh .....

amistre64 · Answer

if something that is larger converges, then the smaller stuff converges by default

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amistre64 · Answer

the sum of x1 to x6 is defined by discrete points along f(x)

if we add up all of f(x) from x1 to x6 we are in effect less than or equal to the sum of all the infinite little parts of f(x)
$$\sum_{i=1}^{6}f(x_i)\le\int_{x_1}^{x_6}f(x)~dx$$

amistre64 · Answer

http://tutorial.math.lamar.edu/Classes/CalcII/IntegralTest.aspx might as well let the experts tell it :)