Question and Answers Forum

All Questions      Topic List

Limits Questions

Previous in All Question      Next in All Question      

Previous in Limits      Next in Limits      

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

f(x)=4x^2 +1 for all x ∈ R x_n =Σ_(k=1) ^n (1/(k^2 +3k+6)) for all n ∈N lim_(n→∞) f(x_n )=...

$${f}\left({x}\right)=\mathrm{4}{x}^{\mathrm{2}} +\mathrm{1}\:\:\mathrm{for}\:\mathrm{all}\:{x}\:\in\:\mathbb{R} \\ $$$${x}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}^{\mathrm{2}} +\mathrm{3}{k}+\mathrm{6}}\:\mathrm{for}\:\mathrm{all}\:{n}\:\in\mathbb{N} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{f}\left({x}_{{n}} \right)=... \\ $$

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

just trying to justify... x_n =Σ_(k=1) ^n (1/(k^2 +3k+6)) x_n =((1/(1^2 +3×1+6))+(1/(2^2 +3×2+6))+(1/(3^2 +3×3+6))+..+(1/(n^2 +3n+6))) =((1/(10))+(1/(16))+(1/(24))+(1/(34))+(1/(46))+...+(1/(n^2 +3n+6))) now ((1/(10))+(1/(16))+(1/(24))+(1/(34))+...+(1/(n^2 +3n+6)))<n now f(x_n )=4x_n ^2 +1 4x_n ^2 +1<4n^2 +1 but lim_(n→∞) (4x_n ^2 +1)<lim_(n→∞) (4n^2 +1) it is not solution but ihe image of my thought

$${just}\:{trying}\:{to}\:{justify}... \\ $$$${x}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}^{\mathrm{2}} +\mathrm{3}{k}+\mathrm{6}} \\ $$$${x}_{{n}} =\left(\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{2}} +\mathrm{3}×\mathrm{1}+\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} +\mathrm{3}×\mathrm{2}+\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} +\mathrm{3}×\mathrm{3}+\mathrm{6}}+..+\frac{\mathrm{1}}{{n}^{\mathrm{2}} +\mathrm{3}{n}+\mathrm{6}}\right) \\ $$$$=\left(\frac{\mathrm{1}}{\mathrm{10}}+\frac{\mathrm{1}}{\mathrm{16}}+\frac{\mathrm{1}}{\mathrm{24}}+\frac{\mathrm{1}}{\mathrm{34}}+\frac{\mathrm{1}}{\mathrm{46}}+...+\frac{\mathrm{1}}{{n}^{\mathrm{2}} +\mathrm{3}{n}+\mathrm{6}}\right) \\ $$$${now} \\ $$$$\left(\frac{\mathrm{1}}{\mathrm{10}}+\frac{\mathrm{1}}{\mathrm{16}}+\frac{\mathrm{1}}{\mathrm{24}}+\frac{\mathrm{1}}{\mathrm{34}}+...+\frac{\mathrm{1}}{{n}^{\mathrm{2}} +\mathrm{3}{n}+\mathrm{6}}\right)<{n} \\ $$$$ \\ $$$${now}\:{f}\left({x}_{{n}} \right)=\mathrm{4}{x}_{{n}} ^{\mathrm{2}} +\mathrm{1} \\ $$$$\mathrm{4}{x}_{{n}} ^{\mathrm{2}} +\mathrm{1}<\mathrm{4}{n}^{\mathrm{2}} +\mathrm{1} \\ $$$${but}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{4}{x}_{{n}} ^{\mathrm{2}} +\mathrm{1}\right)<\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{4}{n}^{\mathrm{2}} +\mathrm{1}\right) \\ $$$${it}\:{is}\:{not}\:{solution}\:{but}\:{ihe}\:{image}\:{of}\:{my}\:{thought} \\ $$$$ \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com