The sequence is uniformly convergent if the value of convergence is uniform, meaning: for any d > 0, there exists N>0 such that the type between the kth partial sum and the shrink is below d for all x, for all ok > N. In different words, the value of convergence (represented by utilising what N is chosen for a given d) does not remember on the value of x. For this sequence the slowest convergence occurs while x is 0, so in case you are able to decide what N works for a given d>0 and for x=0 then I promise which will paintings for all x>0. For x=0 that's the alternating harmonic sequence. It does not converge certainly for an identical reason (extremely) that the harmonic sequence does not converge. that's usually carried out by utilising sequence assessment for any given x - the tail of the harmonic sequence does not converge, and utilising the floor function on x there's a term by utilising term assessment with the tail of the harmonic sequence.
Answers & Comments
Verified answer
Use that
1/(n+x^2) <= 1/n
and
1/n -> 0
at a rate independent of x.
Then use that
1/(n+x^2)
is asymptotic to
1/n
for any fixed x.
The sequence is uniformly convergent if the value of convergence is uniform, meaning: for any d > 0, there exists N>0 such that the type between the kth partial sum and the shrink is below d for all x, for all ok > N. In different words, the value of convergence (represented by utilising what N is chosen for a given d) does not remember on the value of x. For this sequence the slowest convergence occurs while x is 0, so in case you are able to decide what N works for a given d>0 and for x=0 then I promise which will paintings for all x>0. For x=0 that's the alternating harmonic sequence. It does not converge certainly for an identical reason (extremely) that the harmonic sequence does not converge. that's usually carried out by utilising sequence assessment for any given x - the tail of the harmonic sequence does not converge, and utilising the floor function on x there's a term by utilising term assessment with the tail of the harmonic sequence.