By Steven G. Krantz

Do not get me incorrect - Krantz is sweet yet this can be primarily child Rudin - with no the proofs - that is kind of like a bar with no beer.

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**Extra resources for A Handbook of Real Variables: With Applications to Differential Equations and Fourier Analysis**

**Example text**

We call a set U c JR open if, whenever x E U, there is an E > 0 such that (x - E. X + E) cU. In JR. any open set U is the countable union of disjoint open intervals. 3. 3 It may be noied that the union of any number (finite or infinite) of open sets is open. The intersection of finitely many (but. in general. not of infinitely many) open sets is open. 1 Let U = (3,4) U (7,9). Then U is open. To illustrate this point. we take. for instance. 88 E U. 1 and see 0 that (x - E. 98) C S. 2 Closed Sets A set E c JR is called closed if its complement C E is open.

0•.... It is plain that this sequence has no limit. So the series converge. 4 Let Cj = 1/j. 2 Elementary Convergence Tests 1 1 1 1 111 ~=1+-+-+-+-+-+-+- 2 3 4 5 678 = 1+ [~] + [~+~] + U+ ~+~ +4] > 1+ ~ + [~ + ~] + [~ + ~ + ~ + ~] 2448888 111 =1+ 2 + 2 + 2 etc. We see that the sequence SI, S2, ... of partial sums is strictly increasing. and that it has a subsequence that tends to +00. Thus the sequence of partial sums does not tend to a finite limit, and the series diverges. 0 The series in the last example is commonly known as the harmonic series.

The answer is that there are circumI stances when the Ratio Test is much easier to apply than the Root Test. j" j=] is easily studied using the Ratio Test (recall that j! Indeed Cj = 2 j jj! and Cj+ll Cj =j . (j - I) ... 2· I). j = 2 +l j(j + I)!. 2Jjj! We can perform the division to see that IC~;II = j:I' The lim sup of the last expression is O. By the Ratio Test, the series converges. Notice that in this example. while the Root Test applies in principle. it would be difficult to use in practice.