f-x-4x-2-1-for-all-x-R-x-n-k-1-n-1-k-2-3k-6-for-all-n-N-lim-n-f-x-n-

Question Number 56707 by gunawan last updated on 22/Mar/19

f (x) = 4 x^{2} + 1 for all x \in R

x_{n} = \underset{k = 1}{\sum^{n}} \frac{1}{k^{2} + 3 k + 6} for all n \in N

\lim_{n \to \infty} f (x_{n}) = \dots

Answered by tanmay.chaudhury50@gmail.com last updated on 23/Mar/19

j u s t t r y i n g t o j u s t i f y \dots

x_{n} = \underset{k = 1}{\sum^{n}} \frac{1}{k^{2} + 3 k + 6}

x_{n} = (\frac{1}{1^{2} + 3 \times 1 + 6} + \frac{1}{2^{2} + 3 \times 2 + 6} + \frac{1}{3^{2} + 3 \times 3 + 6} + . . + \frac{1}{n^{2} + 3 n + 6})

= (\frac{1}{10} + \frac{1}{16} + \frac{1}{24} + \frac{1}{34} + \frac{1}{46} + \dots + \frac{1}{n^{2} + 3 n + 6})

n o w

(\frac{1}{10} + \frac{1}{16} + \frac{1}{24} + \frac{1}{34} + \dots + \frac{1}{n^{2} + 3 n + 6}) < n

n o w f (x_{n}) = 4 x_{n}^{2} + 1

4 x_{n}^{2} + 1 < 4 n^{2} + 1

b u t \lim_{n \to \infty} (4 x_{n}^{2} + 1) < \lim_{n \to \infty} (4 n^{2} + 1)

i t i s n o t s o l u t i o n b u t i h e i m a g e o f m y t h o u g h t