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Question Number 166940 Answers: 1 Comments: 0

Question Number 166939 Answers: 1 Comments: 0

Question Number 166929 Answers: 1 Comments: 0

A particle of mass 0.25kg vibrates with a period of 2secs. If its greatest displacement is 0.4m. What is its maximum kinetic energy.

$$\:{A}\:{particle}\:{of}\:{mass}\:\mathrm{0}.\mathrm{25}{kg}\:{vibrates} \\ $$$$\:{with}\:{a}\:{period}\:{of}\:\mathrm{2}{secs}.\:{If}\:{its}\: \\ $$$$\:{greatest}\:{displacement}\:{is}\:\mathrm{0}.\mathrm{4}{m}.\:{What} \\ $$$$\:{is}\:{its}\:{maximum}\:{kinetic}\:{energy}. \\ $$

Question Number 166928 Answers: 1 Comments: 0

Question Number 166926 Answers: 3 Comments: 0

lim_(n→∞) (5^n /(n!))=?

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{5}^{{n}} }{{n}!}=? \\ $$

Question Number 166917 Answers: 2 Comments: 4

Question Number 166916 Answers: 0 Comments: 0

Ω= Σ_(n=1) ^∞ (( H_( n) )/(n(n+1))) = −−−−−− Ω = Σ_(n=1) ^∞ −(1/(n+1)) ∫_(0 ) ^( 1) x^( n−1) ln(1−x )dx = ∫_0 ^( 1) {−(1/x^2 )ln(1−x).Σ_(n=1) (x^( n+1) /(n+1))}dx = ∫_0 ^( 1) {((−ln(1−x))/x^( 2) )Σ_(n=2) ^∞ (x^( n) /n)}dx = ∫_0 ^( 1) ((−ln(1−x))/x^( 2) ) {−x +Σ_(n=1) ^∞ (x^( n) /n) }dx = −li_( 2) ( 1) +[ ∫_0 ^( 1) ((ln^( 2) ( 1−x ))/x^( 2) )dx=_(derived) ^(earlier) (π^( 2) /3) ] = −(π^( 2) /6) + (π^( 2) /3) = (( π^( 2) )/6) = ζ (2) ■ m.n

$$ \\ $$$$\:\:\:\:\:\Omega=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:{H}_{\:{n}} }{{n}\left({n}+\mathrm{1}\right)}\:= \\ $$$$\:\:\:\:\:\:\:−−−−−− \\ $$$$\:\:\:\:\:\:\:\Omega\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}−\frac{\mathrm{1}}{{n}+\mathrm{1}}\:\int_{\mathrm{0}\:} ^{\:\mathrm{1}} {x}^{\:{n}−\mathrm{1}} {ln}\left(\mathrm{1}−{x}\:\right){dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \left\{−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }{ln}\left(\mathrm{1}−{x}\right).\underset{{n}=\mathrm{1}} {\sum}\frac{{x}^{\:{n}+\mathrm{1}} }{{n}+\mathrm{1}}\right\}{dx} \\ $$$$\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \left\{\frac{−{ln}\left(\mathrm{1}−{x}\right)}{{x}^{\:\mathrm{2}} }\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{{x}^{\:{n}} }{{n}}\right\}{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{−{ln}\left(\mathrm{1}−{x}\right)}{{x}^{\:\mathrm{2}} }\:\left\{−{x}\:+\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{x}^{\:{n}} }{{n}}\:\right\}{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\:−{li}_{\:\mathrm{2}} \left(\:\mathrm{1}\right)\:+\left[\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}^{\:\mathrm{2}} \left(\:\mathrm{1}−{x}\:\right)}{{x}^{\:\mathrm{2}} }{dx}\underset{{derived}} {\overset{{earlier}} {=}}\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{3}}\:\right] \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\:−\frac{\pi^{\:\mathrm{2}} }{\mathrm{6}}\:+\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{3}}\:=\:\frac{\:\pi^{\:\mathrm{2}} }{\mathrm{6}}\:=\:\zeta\:\left(\mathrm{2}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n} \\ $$

Question Number 166911 Answers: 1 Comments: 0

Question Number 166910 Answers: 1 Comments: 1

given that is prime,proof that (√p) is irrational

$${given}\:{that}\:{is}\:{prime},{proof}\:{that}\:\sqrt{{p}}\:{is}\: \\ $$$${irrational} \\ $$

Question Number 166904 Answers: 1 Comments: 0

A pendulum has a period 2sec. The bob pulled aside a distance 8cm as the amplitude and release. Find the displacement of the bob 0.7sec after it has been released.

$${A}\:{pendulum}\:{has}\:{a}\:{period}\:\mathrm{2}{sec}.\:{The} \\ $$$${bob}\:{pulled}\:{aside}\:{a}\:{distance}\:\mathrm{8}{cm}\:{as} \\ $$$${the}\:{amplitude}\:{and}\:{release}.\:{Find}\:{the} \\ $$$${displacement}\:{of}\:{the}\:{bob}\:\mathrm{0}.\mathrm{7}{sec}\:{after} \\ $$$${it}\:{has}\:{been}\:{released}. \\ $$

Question Number 166881 Answers: 0 Comments: 0