TITLE: Why does $1/x$ diverge?
QUESTION [26 upvotes]: So for the formula $\dfrac {1}{n}$, If you were to add up all $y$ values from $n=1$ to $n=∞$, wouldn't the sum
$$\sum_{n=1}^\infty \frac{1}{n}$$
approach a number because even though you are always adding, aren't you just adding smaller and smaller numbers? Wouldn't this mean that it approached a certain number?

REPLY [13 votes]: As others have explained, the series diverges. But the divergence is very slow, indeed.
See below.
A recent related idea for a first year calculus exercise:
An intelligent robot named Marvin travelled back in time to the moment of the Big Bang 13.7 billion years ago. He started calculating the partial sums of the harmonic series
$$
\frac11+\frac12+\frac13+\frac14+\frac15+\cdots
$$
He added one term to the partial sum per second. Using the estimates leading to the so called integral test answer the following question: As of today, has Marvin's sum reached the value 42?