Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 91667 by Ar Brandon last updated on 02/May/20

find the limits of 1. u_n =Σ_(k=1) ^n (n/(n^2 +k^2 )) 2. v_n =Σ_(k=1) ^n (1/(√(n^2 +2kn)))

$$\mathrm{find}\:\mathrm{the}\:\mathrm{limits}\:\mathrm{of}\: \\ $$$$\mathrm{1}.\:\mathrm{u}_{\mathrm{n}} =\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{n}}{\mathrm{n}^{\mathrm{2}} +\mathrm{k}^{\mathrm{2}} }\:\:\: \\ $$$$\mathrm{2}.\:\mathrm{v}_{\mathrm{n}} =\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\sqrt{\mathrm{n}^{\mathrm{2}} +\mathrm{2kn}}} \\ $$

Commented by Prithwish Sen 1 last updated on 02/May/20

1.lim_(n→∞) U_n =lim_(n→∞) (1/n)Σ_(k=1) ^n (1/(1+((k/n))^2 )) = ∫_0 ^1 (dx/(1+x^2 )) = tan^(−1) (π/4) 2.lim_(n→∞) V_n = lim_(n→∞) (1/n)Σ_(k=1) ^n (1/(√(1+2((k/n))))) = ∫_0 ^1 (dx/(√(1+2x)))=[(√(1+2x))]_0 ^1 = (√3)−1

$$\mathrm{1}.\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}U}_{\mathrm{n}} =\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{n}}\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{1}+\left(\frac{\mathrm{k}}{\mathrm{n}}\right)^{\mathrm{2}} }\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{dx}}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\:=\:\mathrm{tan}^{−\mathrm{1}} \frac{\pi}{\mathrm{4}} \\ $$$$\mathrm{2}.\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{V}_{\mathrm{n}} =\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{n}}\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\sqrt{\mathrm{1}+\mathrm{2}\left(\frac{\mathrm{k}}{\mathrm{n}}\right)}}\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{dx}}{\sqrt{\mathrm{1}+\mathrm{2x}}}=\left[\sqrt{\mathrm{1}+\mathrm{2x}}\right]_{\mathrm{0}} ^{\mathrm{1}} \:=\:\sqrt{\mathrm{3}}−\mathrm{1} \\ $$

Commented by Ar Brandon last updated on 02/May/20

Thanks What′s the relationship between the Σ and ∫ please?

$$\mathrm{Thanks} \\ $$$$\mathrm{What}'\mathrm{s}\:\mathrm{the}\:\mathrm{relationship}\:\mathrm{between}\:\mathrm{the}\:\Sigma\:\mathrm{and}\:\int \\ $$$$\mathrm{please}? \\ $$

Commented by Ar Brandon last updated on 02/May/20

What if it was from (1/n) to 1 instead of 0 to 1?

$$\mathrm{What}\:\mathrm{if}\:\mathrm{it}\:\mathrm{was}\:\mathrm{from}\:\frac{\mathrm{1}}{\mathrm{n}}\:\mathrm{to}\:\mathrm{1}\: \\ $$$$\mathrm{instead}\:\mathrm{of}\:\mathrm{0}\:\mathrm{to}\:\mathrm{1}? \\ $$

Commented by Prithwish Sen 1 last updated on 02/May/20

when you add some discrete entity for a very large number you put Σ like if you add all the natural number from 1 to ∞ you put Σ_(n=1) ^∞ n but if you add an continuos entity such as time which takes all the values in between the natural numbers you use ∫

$$\mathrm{when}\:\mathrm{you}\:\mathrm{add}\:\mathrm{some}\:\mathrm{discrete}\:\mathrm{entity}\:\mathrm{for}\:\mathrm{a}\:\mathrm{very}\:\mathrm{large} \\ $$$$\mathrm{number}\:\mathrm{you}\:\mathrm{put}\:\Sigma \\ $$$$\mathrm{like}\:\mathrm{if}\:\mathrm{you}\:\mathrm{add}\:\mathrm{all}\:\mathrm{the}\:\mathrm{natural}\:\mathrm{number}\:\mathrm{from}\:\mathrm{1}\:\mathrm{to} \\ $$$$\infty\:\mathrm{you}\:\mathrm{put}\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\mathrm{n}\:\mathrm{but}\:\mathrm{if}\:\mathrm{you}\:\mathrm{add}\:\mathrm{an}\:\mathrm{continuos}\:\mathrm{entity} \\ $$$$\mathrm{such}\:\mathrm{as}\:\mathrm{time}\:\mathrm{which}\:\mathrm{takes}\:\mathrm{all}\:\mathrm{the}\:\mathrm{values}\:\mathrm{in}\:\mathrm{between} \\ $$$$\mathrm{the}\:\mathrm{natural}\:\mathrm{numbers}\:\mathrm{you}\:\mathrm{use}\:\int \\ $$

Commented by Ar Brandon last updated on 02/May/20

OK got it! Thanks

$$\mathrm{OK}\:\mathrm{got}\:\mathrm{it}!\:\mathrm{Thanks} \\ $$

Commented by mathmax by abdo last updated on 02/May/20

1) u_n =(1/n)Σ_(k=1) ^n (1/(1+((k/n))^2 )) ⇒u_n is aRieman sum and lim_(n→∞) u_n =∫_0 ^1 (dx/(1+x^2 )) =arctan(1) =(π/4)

$$\left.\mathrm{1}\right)\:{u}_{{n}} =\frac{\mathrm{1}}{{n}}\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{\mathrm{1}+\left(\frac{{k}}{{n}}\right)^{\mathrm{2}} }\:\Rightarrow{u}_{{n}} {is}\:{aRieman}\:{sum}\:{and}\: \\ $$$${lim}_{{n}\rightarrow\infty} \:{u}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}} }\:={arctan}\left(\mathrm{1}\right)\:=\frac{\pi}{\mathrm{4}} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com