Help with math probs plzzzz?

Question

1). Find the limit Lim as n approaches infinity of the series k=1 and it reaches n of sqrt(k/n) * (1/n)=???

2). if dy/dx =y-2x+3 where y =f(x) is te solution to the equation and f(2) =5 Using Euler's method starting at x sub 0 =2 with step size delta x=.5, what is the approximation for f(3)?

3). Lim as x approaches 0 of (cos x-e^x)/(ln(1+x)) =????

4). The first four terms of the taylor expansion for f(x) about x=3 are 5-((x-3)/4)-(7(x-3)^2)/3+(9(x-3)^3)/2...what is the value of f '' (3)?

Answer 1

3)
Lim as x approaches 0 of (cos x-e^x)/(ln(1+x))
=0/0 indefinite
take derivative of numerator and denominator

Lim as x approaches 0
(-sinx-e^x)/ (1/(1+x))
=(-0-1)/(1+0)
=-1/1
=-1

Answer 2

(1) Remember that the integral is an infinite sum.. this is simply equal to integral from 0 to 1 of sqrt(x)
int0to1 (sqrt(x) dx) = 2/3 * x^(3/2) evaluated 0 to 1
= 2/3

(2) (2,5) m=4 ===> (2.5 , 6) m=4 ===> (3,10)

f(3) approximately 10


(3) L'Hopital's rule

lim x => 0 (-sinx -e^x) / (1/(1+x) ) = -1/1 = -1

(4)

f ' (x) = -1/4 - 14 (x-3)/3 + ....
f " (x) = -14/3 + ...... all terms will goto zero when plug in x=3
f " (x) = -14/3