Ramanujan's reasoning is analysed in detail by Berndt, in his discussion of chapters 6,7,8 of Ramanujan's notebooks. Ramanujan starts from the Euler-MacLaurin summation formula, $$\sum_{k=1}^x f(k)=C +\int_1^x f(t)\,dt+\tfrac{1}{2}f(x)+\sum_{k\geq 1}\frac{B_{2k}}{(2k)!}\frac{d^{2k-1}}{dx^{2k-1}}f(x),$$ and focuses on the constant $C$, which he uses to find the asymptotic expansion of $\sum_{k=1}^x f(k)$ as $x\rightarrow\infty$. This approach was made somewhat more rigorous by Hardy.

In this way Ramanujan obtained the notorious identities $\sum 1 =-\tfrac{1}{2}$ and $\sum k = -\tfrac{1}{12}$. For a $p$-adic interpretation of the latter sum, see this MO post.

Ramanujan's reasoning is analysed in detail by Berndt, in his discussion of chapter 6 of Ramanujan's notebooks. Ramanujan starts from the Euler-MacLaurin summation formula, $$\sum_{k=1}^x f(k)=C +\int_1^x f(t)\,dt+\tfrac{1}{2}f(x)+\sum_{k\geq 1}\frac{B_{2k}}{(2k)!}\frac{d^{2k-1}}{dx^{2k-1}}f(x),$$ and focuses on the constant $C$, which he uses to find the asymptotic expansion of $\sum_{k=1}^x f(k)$ as $x\rightarrow\infty$. This approach was made somewhat more rigorous by Hardy.

In this way Ramanujan obtained the notorious identities $\sum 1 =-\tfrac{1}{2}$ and $\sum k = -\tfrac{1}{12}$. For a $p$-adic interpretation of the latter sum, see this MO post.

Ramanujan's reasoning is analysed in detail by Berndt, in his discussion of chapters 6,7,8 of Ramanujan's notebooks. Ramanujan starts from the Euler-MacLaurin summation formula, $$\sum_{k=1}^x f(k)=C +\int_1^x f(t)\,dt+\tfrac{1}{2}f(x)+\sum_{k\geq 1}\frac{B_{2k}}{(2k)!}\frac{d^{2k-1}}{dx^{2k-1}}f(x),$$ and focuses on the constant $C$, which he uses to find the asymptotic expansion of $\sum_{k=1}^x f(k)$ as $x\rightarrow\infty$. This approach was made somewhat more rigorous by Hardy.

In this way Ramanujan obtained the notorious identities $\sum 1 =-\tfrac{1}{2}$ and $\sum k = -\tfrac{1}{12}$.

Ramanujan's reasoning is analysed in detail by Berndt, in his discussion of chapters 6,7,8 of Ramanujan's notebooks. Ramanujan starts from the Euler-MacLaurin summation formula, $$\sum_{k=1}^x f(k)=C +\int_1^x f(t)\,dt+\tfrac{1}{2}f(x)+\sum_{k\geq 1}\frac{B_{2k}}{(2k)!}\frac{d^{2k-1}}{dx^{2k-1}}f(x),$$ and focuses on the constant $C$, which he uses to find the asymptotic expansion of $\sum_{k=1}^x f(k)$ as $x\rightarrow\infty$. This approach was made somewhat more rigorous by Hardy.

In this way Ramanujan obtained the notorious identities $\sum 1 =-\tfrac{1}{2}$ and $\sum k = -\tfrac{1}{12}$. For a $p$-adic interpretation of the latter sum, see this MO post.