Question

mark true or false

8. If the seriesh converges absolutely then the series sin (kz) converges uniformly on R. 9. There exists a polynomial f such that its Taylor series centered at 1 does not converge to f. 10. If a sequence of functions (fr) converges uniformly on sets D and E, then it converges uniformly on the set DUE

Answer 1

8. TRUE

Defⁿ :- An infinite series of real numbers is said to be absolutely convergent if the sum of absolute values of the summands is finite.

Defⁿ :- A series of functions
Ση(1)
is said to converge uniformly on
E CR
if te sequence of partial sums $\{s_n\}$ defined by
ML]
converges uniformly to a function on E. i.e. if for each $\epsilon> 0$ , an integer N can be found such that
sn() - (2) <eyn> N and ZEE
.

An important test to check for uniform convergence of a series of function to be uniformly convergent is WEIERSTRASS' M TEST. It says that

If
E CR
& $\sum f_n$ be a seies of function on E. Let $\{M_n\}$ be a sequence of positive real numbers such that
Vr EE, Fn(1) <Mn VnEN.
If the sreies $\sum M_n$ is convergent then the series $\sum f_n$ is uniformly and absolutely convergent on E.

het for (x) = ansin (nn) N=1,2, .-- Since I Sin (non) I sI +2 GR So If, (a) / <1 anl #ngor & tecer. - Now it is given that E are is absolutely convergent So the sequence {iande comerges. There fone by Weerstras not an test elu) convergent #neo. MEIN (9) nal