I am a beginner in calculus and I want to know what is the difference between sum and integral. More specifically I came across this example:
Compare $$\sum^\infty_1\frac1x\space \text{and} \space \int_1^\infty\frac1xdx$$
It would be really helpful if someone explains this to me. I want to know the difference between the two.
Thanks for any help!!
Best Answer
Basically, an integral adds up infinitely small pieces, whereas a sum adds up distinct pieces. $$ \int_1^{\infty} xdx\, $$
Will add up all of the area under f(x)=x, whereas $$ \sum_{x=1}^{\infty} x\, $$ Will add up each value for x from whatever you start, until you stop.
For $f(x)=1/x$ this is the difference between the area under 1/x, and the series $1 + 1/2 + 1/3 + 1/4 +...+ 1/n $ (the harmonic series).
Both of these values diverge.