Math 5C Discussion Problems 3

Transcription

1 Math 5C Discussio Problems 3 Power Series 1. Fid the radius ad iterval of covergece for the followig power series. a) 5 x / b) x / l c) 1) +3 x +1 /!) d) x /1 + ) e) x /l ). Assume that a 3 coverges absolutely. What ca you say about the covergece of each of the followig? a) a 4 b) a c) a 3) d) a 3. Assume that a 4 coverges coditioally. What ca you say about the covergece of each of the followig? a) a 5 b) a c) a 3) d) a 4. Give that fx) = a x coverges i a eighborhood of zero, what is f )? 5. Which of the followig are true for x ear zero? a) x = Ox 3 ) b) x 3 = Ox ) c) x + x 4 = Ox 3 ) 6. Assume that y = x + x! + x3 3! +. Fid the first two ozero terms of a series expasio of y i terms of x. Write the remaider usig Ox ) otatio. 7. Assume that z = y y + y3 3 y4 4 + y = x + x3 3! + x5 5! + Fid the first two ozero terms of a series expasio of z i terms of x. Write the remaider usig Ox ) otatio.

2 Taylor Series 1. Usig the Taylor series of exp, si ad cos, show that e iθ = cos θ + i si θ.. Prove these idetities for arbitrary real umbers x. a) sihix) = i si x b) coshix) = cos x 3. Use a Taylor series to approxmate the followig. a) e.3 with a error less tha.1. b) si.1) with a error less tha 1 1 c) d) 1 1 e x dx with 1 decimal place accuracy error less tha.5) 1 cos x x 4. Evaluate the followig its. si x 3 a) x 3 b) e1/ 1) c) cossi x) cos x arctax 4 ) e x cos x 1 d) e) + x x x 1 x l x e si x 1 x f) x g) arcta h) ) e dx with decimal place accuracy error less tha.5) ) 1 si 5. Evaluate the followig series. a) ! + 1 3! + 1 4! + b) π π3 3! + π5 5! π7 7! + c) ! ! ! + d) e) )) 1 f)

3 Fourier Series 1. For each of the followig, compute a) fx) = cos x + cos 1x + si 3x b) fx) = si x c) fx) = cos 3 x d) fx) = si x cos x fx) si x dx. No itegrals eed be computed.. For each of the followig π-periodic fuctios, compute the Fourier series. { 1, < x < a) fx) = 1, < x < π b) fx) = x whe < x < π c) fx) = x whe < x < π d) fx) = e x whe < x < π 3. Show that the product of eve fuctios is eve, the product of odd fuctios is odd, ad the product of a eve ad a odd fuctio is odd. 4. a) Suppose that f is a eve fuctio. Show that b = for all. b) Suppose that f is a odd fuctio. Show that a = for all. 5. Let fx) = x for < x < π, exteded periodically. a) Is f eve or odd? b) Show that the Fourier series of f is fx) 4π cos x =1 c) Use the Fourier series to evaluate =1 by settig x =. d) Use the Fourier series to evaluate =1 1). e) Use Parseval s idetity to evaluate = ) π si x 6. Use Parseval s idetity to deduce the followig fact. Give a π-periodic fuctio f with fx) dx <, it follows that fx) cos x dx = fx) si x dx =. 7. From the expasio l si x ) = use Parseval s idetity to evaluate the itegral =1 cos x, < x < π, l si x ) dx.

4 Series: Miscellay 1. We kow that 1 + 1/) e whe. Usig series, fid the umber a that satifies ) = e + a ) 1 + O for large. Show that, for large itegers, Use this to evaluate siπ!e). siπ!e) = π O ) 3. Use what you kow about approximatig fuctios to determie coveregece of these series. a) )) 1 1 cos b) )) 1 1 si c) )) 1 log si use the previous part ad it compariso) d) 1 e 1/α) the aswer depeds o α) 4. Prove that for 1 < x < 1, ) x = = =1 x 1 5. Not for the fait of heart!) About years ago, reowed mathematical physicist V.I. Arold declared that moder studets of mathematics are poorly traied. Sobolev embeddig theorems ad abelia category theory are othig if you ca t compute the average value of si 1 x withi 1% i uder five miutes or so he claims). Arold the posted a mathematical trivium, that is, 97 problems ay math studet should be able to solve with miimal effort. Here s problem umber from the list: evaluate si ta x ta si x arcsi arcta x arcta arcsi x. 6. See above problem descriptio) For those who ca t get eough Arold, here s trivium umber 18. Evaluate j exp x j x k dx 1 dx dx. R j=1 k=1

5 Harmoic Fuctios 1. Which of these fuctios are harmoic o R? a) fx, y) = e x si y b) fx, y) = x y c) fx, y) = x + y d) fx, y) = lx + y ). Give a example of harmoic fuctios f ad g so that fg is ot harmoic. 3. Assume that fx, y) is harmoic o R. Show that zfx, y) is harmoic o R Assume the maximum priciple is true. Show that the uiqueess theorem follows. Hit: f g 5. Give a reaso why F) f). 6. Let f be harmoic o R 3 ad B be a ball. Use the divergece theorem to show that f f da = f dv. 7. Use the previous problem to prove the followig: If f is harmoic ad f = o the surface of a ball, the f = everywhere iside the ball. 8. Use the uiqueess for harmoic fuctios to prove the followig. Suppose smooth scalar fuctios f ad g o R 3 are equal o the surface of a ball B. Suppose that f = g; show that f = g iside B. 9. Use the fact that lx + y ) is harmoic i R except at the origi) to aswer the followig: Fid a fuctio fx, y) which is equal to 1 o the circle x + y = 1, equal to whe x + y = 4, ad is harmoic whe 1 < x + y < 4. What are the odds that someoe else got the same aswer? 1. Fid a harmoic fuctio fx, y, z) which is idetically 1 o surface of the sphere x + y + z = Assume that f ad g are harmoic fuctios i R 3 ad B is a ball i R 3. Use the divergece theorem to show that f g da = g f da. 1. Let B be the uit ball i R 3 cetered at 1, 1, 1) ad defie fx, y, z) = z lx + y ). Evaluate f dσ. 13. Let B be the uit ball i R 3 cetered at 1,, ) ad defie fx, y, z) = x ly + z ). Explaiig your reasoig, evaluate f dσ. B