One must make an effort to remember correct statements of definitions and results.
The process of summing an infinite series is a process that usually (there are several ways) involves limits. Limits do not always behave as one might expect.
We have studied an example of a sequence {fn}, n = 1, 2, 3, ... where the limit of the sequence fn is 0 but the limit of the sequence of integrals over the interval [0, 1] is 1/2.