- Suppose 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 sequence converges uniformly to 0 on [0,∞)? Now replace [0,∞) with. What can be said in this case assuming pointwise convergence still holds?
- If andare sequences of functions defined on D which converge uniformly, show that if a,b are constants, then af
_{n}+ bg_{n}also converges uniformly. If there exists a constant, M such that,< M for all n and for all x ∈ D, showconverges uniformly. Let f_{n}≡ 1∕x for x ∈and let g_{n}≡∕n. Showconverges uniformly onandconverges uniformly butfails to converge uniformly. - Show that if x > 0,∑
_{k=0}^{∞}converges uniformly on any interval of finite length. - Let x ≥ 0 and consider the sequence . Show this is an increasing sequence and is bounded above by ∑
_{k=0}^{∞}. - Suppose f is a continuous function and
for n = 0,1,2,3

. Show that f= 0 for all x. Hint: You might use the Weierstrass approximation theorem. - Show for every x,y real, ∑
_{k=0}^{∞}converges and equals - Consider the series ∑
_{n=0}^{∞}^{n}. Show this series converges uniformly on any interval of the form. - Formulate a theorem for a series of functions which will allow you to conclude the infinite series is uniformly continuous based on reasonable assumptions about the functions in the sum.
- Find an example of a sequence of continuous functions such that each function is
nonnegative and each function has a maximum value equal to 1 but the sequence of
functions converges to 0 pointwise on .
- Suppose is a sequence of real valued functions which converges uniformly to a continuous function f. Can it be concluded the functions f
_{n}are continuous? Explain. - Let hbe a bounded continuous function. Show the function f= ∑
_{n=1}^{∞}is continuous. - Let S be a any countable subset of ℝ. This means S is actually the set of terms of a
sequence. That is S =
_{n=1}^{∞}. Show there exists a function f defined on ℝ which is discontinuous at every point of S but continuous everywhere else. Hint: This is real easy if you do the right thing. It involves Theorem 10.7.4 and the Weierstrass M test. - By Theorem 10.8.3 there exists a sequence of polynomials converging uniformly to
f=on the interval. Show there exists a sequence of polynomials,converging uniformly to f onwhich has the additional property that for all n,p
_{n}= 0 . - If f is any continuous function defined on , show there exists a series of the form ∑
_{k=1}^{∞}p_{k}, where each p_{k}is a polynomial, which converges uniformly to f on. Hint: You should use the Weierstrass approximation theorem to obtain a sequence of polynomials. Then arrange it so the limit of this sequence is an infinite sum. - Sometimes a series may converge uniformly without the Weierstrass M test being
applicable. Show
converges uniformly on

but does not converge absolutely for any x ∈ ℝ. To do this, it might help to use the partial summation formula, 10.6. - This problem outlines an approach to Stirling’s formula which is found in [32] and
[7]. From the above problems, Γ= n! for n ≥ 0. Consider more generally Γfor x > 0. It equals ∫
_{0}^{∞}e^{−t}t^{x}dt. Change variables letting t = xto obtainNext let h

be such that h= 1 andShow that the thing which works is h

=. Use L’Hospital’s rule to verify that the limit of has u → 0 is 1. The graph of h is illustrated in the following picture. Verify that its graph is like this, with an asymptote at u = −1 decreasing and equal to 1 at 0 and converging to 0 as u →∞.Next change the variables again letting u = s

. This yieldsConsider the integrand in the above. Using the description of h in the above graph, verify that

and that this convergence is uniform on any interval of the form

. So that all makes sense, you should have x large enough thatTo verify this convergence, explain why it suffices to verify the uniform convergence of h

to 1 on this interval. Use the graph to observe why this is so. Now consider and explain the following inequality valid for> Aand x large.First consider s < 0 and the first integral on the right of the inequality. For s < 0, you have h

> 1. See the above graph. Thus the integrand is no larger than e^{−s2 }. Now consider the fourth integralThus, this integrand is dominated by

expfor s > 0 and large x. Explain why this is so. Therefore,Explain why this last integral equals

. See Problem 35. This yields a general Stirling’s formula, - To show you the power of Stirling’s formula, find whether the series
converges. The ratio test falls flat but you can try it if you like. Now explain why, if n is large enough,

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