Solve the equation dx
dy
​
+y=e −x
,y∣ x=0
​
=1. 2. Show the series ∑ n=1
[infinity]
​
(−1) n−1
n
n
​
+1
​
is convergent or not? If it is convergent, show it is absolute convergence or conditional convergence? 3. Show the interval of convergence and the sum function of ∑ n=0
[infinity]
​
3 n
(n+1)x n
​
. 4. Expand the function f(x)=e 2x
(e x
+1) into the power series. 5. Show the general solution of y ′′
=y ′
+x. 6. If y=f(x) is defined by { x=t−arctant
y=ln(1+t 2
)
​
, show dx 2
d 2
y
​
.