Does the series 1/n converge to zero? I know that it is a harmonic series, but shouldn't it converge because as x gets really large to infinity, 1/n becomes zero??
Look at this clever video, remarkable!
https://www.youtube.com/watch?v=_Ui5_-lIK34
Oh, got it. Thank you!!
Actually Sal Khan does a really nice job of this:
https://www.youtube.com/watch?v=4yyLfrsSXQQ
Just watched it and it makes more sense. Thank you so much!
The series 1/n, also known as the harmonic series, does not converge to zero. Although the individual terms of the series, 1/n, get arbitrarily close to zero as n approaches infinity, the sum of the terms does not approach a finite value.
To understand why the harmonic series does not converge, we can use the integral test. The integral test states that if a function f(x) is positive, continuous, and decreasing for x ≥ 1, and the series ∑[n=1 to ∞] f(n) is convergent, then the integral ∫[1 to ∞] f(x) dx is also convergent.
Applying the integral test to the harmonic series, we can consider the function f(x) = 1/x. This function meets the conditions of the integral test since it is positive, continuous, and decreasing for x ≥ 1.
To evaluate the integral, we integrate f(x) from 1 to ∞:
∫[1 to ∞] 1/x dx = ln(x) | [1 to ∞] = ln(∞) - ln(1) = ∞.
Since the integral from 1 to ∞ of f(x) diverges (equals infinity), the series 1/n also diverges. Therefore, the harmonic series does not converge to zero.
It is important to note that even though the individual terms 1/n tend towards zero as n increases, the sum of those terms diverges because the terms decrease in value too slowly to approach a finite value.