Infinite series which should have a finite answer

Question

In infinite series we say that some series coverage and some diverge but the series. 1 + 1/2 + 1/3 + 1/4 •••• why we say that it diverges and has no finite answer , as it looks converging to me and also that 1+2+3+4•••••. Is also diverging but has finite answer

Answer 1

1 + 1/2 + 1/3 + 1/4 •••• why we say that it diverges and has no finite answer , as it looks converging to me

Just because it looks convergent doesn't mean that it is. To see that it's divergent, just add some parentheses: $$(1)+({1\over 2})+({1\over 3}+{1\over 4})+({1\over 5}+{1\over 6}+{1\over 7}+{1\over 8})+...$$ It's easy to check that proceeding in this way, the harmonic series is the sum of infinitely many "blocks" each of which sums to $\ge{1\over 2}$, hence it has to diverge. It diverges very slowly, but it does diverge.

1+2+3+4•••••. Is also diverging but has finite answer

What? It pretty obviously does not have a finite answer. Maybe you're referring to the sometimes-encountered claim "$1+2+3+...=-{1\over 12}$," in which case it's important to point out that that's wildly misleading without context.

Answer 2

There's more than one way to define the sum of infinitely many terms. Usually, mathematicians define it as the $n\to\infty$ limit of the sum of the first $n$ terms. In this sense, both series become infinite. (This is obvious for the integers; for the fractions, which form the so-called harmonic series, see here, or pick your favourite among many proofs).

A different definition of the sum is needed if you want $1+2+3+\cdots$ to be finite; you might, for example, use zeta function generalisation to get a value of $-\frac{1}{12}$. (Other methods will say the same, or that the sum is infinite.) By contrast, the harmonic series resists this make-it-finite method (but not all of them).

As for $1+1+1+\cdots$, a similar treatment to the above would obtain $-\frac12$; but, again, this isn't the usual partial-sums limit, which would of course be $\infty$.

Finally, other finite answers are possible with suitable summation methods, as @reuns has noted. So, by taking the right linear combination of two methods, we can get any answer we want. Therefore, we have to know which definition of an "infinite sum" we have in mind when we assert anything.