6.9. EXERCISES 169

Proof: Let z ∈ D. Then letting m < n,∣∣∣∣∣ n

∑k=1

fk (z)−m

∑k=1

fk (z)

∣∣∣∣∣≤ n

∑k=m+1

| fk (z)| ≤∞

∑k=m+1

Mk < ε

whenever m is large enough because of the assumption that ∑∞n=1 Mn converges. Therefore,

the sequence of partial sums is uniformly Cauchy on D and therefore, converges uniformlyto ∑

∞k=1 fk on D.

Theorem 6.8.4 If { fn} is a sequence of functions defined on D which are continuousat z and ∑

∞k=1 fk converges uniformly, then the function ∑

∞k=1 fk must also be continuous at

z.

Proof: This follows from Theorem 4.9.3 applied to the sequence of partial sums of theabove series which is assumed to converge uniformly to the function ∑

∞k=1 fk.

6.9 Exercises1. Suppose { fn} is a sequence of decreasing positive functions defined on [0,∞) which

converges pointwise to 0 for every x ∈ [0,∞). Can it be concluded that this sequenceconverges uniformly to 0 on [0,∞)? Now replace [0,∞) with (0,∞) . What can be saidin this case assuming pointwise convergence still holds?

2. If { fn} and {gn} are sequences of functions defined on D which converge uniformly,show that if a,b are constants, then a fn+bgn also converges uniformly. If there existsa constant, M such that | fn (x)| , |gn (x)|< M for all n and for all x ∈ D, show { fngn}converges uniformly. Let fn (x)≡ 1/x for x ∈ (0,1) and let gn (x)≡ (n−1)/n. Show{ fn} converges uniformly on (0,1) and {gn} converges uniformly but { fngn} fails toconverge uniformly.

3. Show that if x > 0,∑∞k=0

xk

k! converges uniformly on any interval of finite length.

4. Let x≥ 0 and consider the sequence{(

1+ xn

)n}. Show this is an increasing sequence

and is bounded above by ∑∞k=0

xk

k! .

5. Show for every x,y real, ∑∞k=0

(x+y)k

k! converges and equals(∞

∑k=0

yk

k!

)(∞

∑k=0

xk

k!

)

6. Consider the series ∑∞n=0 (−1)n x2n+1

(2n+1)! . Show this series converges uniformly on anyinterval of the form [−M,M] .

7. Formulate a theorem for a series of functions which will allow you to conclude theinfinite series is uniformly continuous based on reasonable assumptions about thefunctions in the sum.

8. Find an example of a sequence of continuous functions such that each function isnonnegative and each function has a maximum value equal to 1 but the sequence offunctions converges to 0 pointwise on (0,∞) .