Question

Evaluate $\displaystyle \rm \lim _ {n\to \infty } \dfrac{1}{1 + {n}^{2} } + \dfrac{2}{2 + {n}^{2} } + \cdots + \dfrac{n}{n + {n}^{2} }$
​

Answer 1

EXPLANATION.

$\sf \implies \displaystyle \lim_{n \to \infty} \bigg[ \frac{1}{1 + n^{2} } + \frac{2}{2 + n^{2} } + . . . . . + \frac{n}{n + n^{2} } \bigg]$

As we know that,

Concept of sandwich theorem.

⇒ If g(x) ≤ f(x) ≤ h(x)  ∨ x ∈(a - h, a + h).

$\sf \implies \displaystyle \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$

$\sf \implies \displaystyle \lim_{x \to a} f(x) = L$

Using this concept in the equation, we get.

$\sf \implies \displaystyle \lim_{n \to \infty} g(n) = \frac{1}{n + n^{2} } + \frac{2}{n + n^{2} } + . . . . . + \frac{n}{n + n^{2} }$

$\sf \implies \displaystyle \lim_{n \to \infty} g(n) = \frac{n(n + 1)}{2(n + n^{2} )}$

$\sf \implies \displaystyle g(n) = \frac{(n^{2} + n)}{2(n + n^{2} )} = \frac{1}{2}$

$\sf \implies \displaystyle \lim_{n \to \infty} h(n) = \frac{1}{1 + n^{2} } + \frac{2}{1 + n^{2} } + . . . . . + \frac{n}{1 + n^{2} }$

$\sf \implies \displaystyle \lim_{n \to \infty} h(n) = \frac{n(n + 1)}{2(1 + n^{2} )}$

$\sf \implies \displaystyle \lim_{n \to \infty} h(n) = \frac{(n^{2}+ n) }{2(1 + n^{2}) }$

$\sf \implies \displaystyle \lim_{n \to \infty} h(n) = \frac{n^{2}(1 + 1/n) }{2n^{2}(1/n^{2} + 1) } = \frac{1}{2}$

As we can see that,

$\sf \implies \displaystyle \lim_{n \to \infty} g(n) = \lim_{n \to \infty} h(n) = \frac{1}{2}$

$\sf \implies \displaystyle \boxed{ \lim_{n \to \infty} f(n) = \frac{1}{2} }$

Answer 2

$P_n = \dfrac{1}{1 + {n}^{2} } + \dfrac{2}{2 + {n}^{2} } + \cdots + \dfrac{n}{n + {n}^{2} }$

$P_n < \dfrac{1}{1 + {n}^{2} } + \dfrac{2}{2 + {n}^{2} } + \cdots + \dfrac{n}{n + {n}^{2} }$

$: \longmapsto \dfrac{1}{1 + {n}^{2} } (1 + 2 + 3 + \cdots + n)$

$: \longmapsto \dfrac{n(n + 1)}{2 \big(1 + {n}^{2} \big)}$

Also,

$P_n > \dfrac{1}{1 + {n}^{2} } + \dfrac{2}{2 + {n}^{2} } + \dfrac{3}{3 + {n}^{2} } \cdots + \dfrac{n}{n + {n}^{2} }$

$: \longmapsto \dfrac{n(n + 1)}{2 \big(n + {n}^{2} \big) } < P_n < \dfrac{n(n + 1)}{2 \big(1 + {n}^{2} \big)}$

$: \longmapsto \displaystyle \lim_{n \to \infty } \dfrac{n(n + 1)}{2 \big(n + {n}^{2} \big)} < \lim_{x \to \infty }P_n < \lim_{x \to \infty }{\dfrac{n(n + 1)}{2 \big(1 + {n}^{2} \big)}}$

$: \longmapsto \displaystyle \lim_{n \to \infty } \dfrac{1 \bigg(1 + \dfrac{1}{n} \bigg)}{2 \bigg( \dfrac{1}{n} + 1 \bigg)} < \lim_{n \to \infty }P_n < \lim_{n \to \infty }\dfrac{1 \bigg(1 + \dfrac{1}{n} \bigg)}{2 \bigg( \dfrac{1}{n^{2} } + 1 \bigg)}$

$: \longmapsto \displaystyle \dfrac{1}{2} < \lim_{n \to \infty }P_n < \dfrac{1}{2}$

$: \longmapsto \displaystyle \lim_{n \to \infty }P_n = \dfrac{1}{2}$