推 ss355227:水喔 04/14 00:18
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→ sky2857: 水^4喔 04/16 16:32
1/x
1.(a)Find lim(1+x)
x→0+
-kx
(b)Let k>0. Show that the function f(x)=ke x>=0 is a probability density
=0 x<0
function.
2. Show that
k 2 3
x ∞ x x x
(a)e =Σ-----= 1 + x + --- + ---+..... for all real x
k=0 k! 2! 3!
∞ 1 2k
(b)coshx= Σ------- x for all real x
k=0 (2k)!
3.(a)Expand sinx and cosx in powers of x-a
(b)Show that both of above series are absolutely convergent for all real x
4.Find the interval of convergence of the following power series.
k
(a)Σ-----x^k
10^k
k^3
(b)Σ -----(x-4)^k
e^k
sinx-x
5. (a) Evaluate the limit lim----------- by (i)L'Hopital rule, (ii)by using
x→0 x^3
power series
(b)Find a power series representation for the improper intergral:
x sinht
∫--------dt
0 t
--
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