Answer to Question #60680 in Real Analysis for esita

Question

8. a) Check whether the following function has a mean value in the interval ]5,2[




≤ ≤
≤ <
=
3 if 3 5
1 if 2 3
( )
x
x
f x
Does this contradict the mean value theorem? Justify. (3)
b) Find the limit as n → ∞ , of the sum

n n
n
n
n
n
n
4
1
3 1 3 2 3 3
2 2 2 2 2 2
+ +
+
+
+
+
+
L . (4)
5
c) Apply Weierstrass M -test to show that the series ∑ +
4 4
10
n x
converges uniformly
for all x ∈ R .