By Steven G. Krantz

Do not get me flawed - Krantz is sweet yet this can be primarily child Rudin - with out the proofs - that's type of like a bar with no beer.

**Read or Download A Handbook of Real Variables: With Applications to Differential Equations and Fourier Analysis PDF**

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

**Example text**

The series consisting of all the negative terms of the series will diverge. Thus we construct the desired rearrangement by using the following steps: • First select just enough positive terms to obtain a partial sum that is greater than 5. • Then add on enough negative terms so that the partial sum falls below 5. • Now add on enough positive terms so that the partial sum once again exceeds 5. • Again add on enough negative terms so that the partial sum falls below 5. Now continue in this fashion.

4 Root Test for Divergence Consider the series If then the series diverges. 5 Ratio Test for Divergence Consider the series If there is an N > 0 such that IC~;II::: 1 , forallj::: N then the series diverges. In both the Root Test and the Ratio Test, if the lim sup is equal to 1, then no conclusion is possible. lim j l2/3 J-+OO = +00. We conclude. 4. that the series diverges. J/J·]2 I = j-+oo =I. J The Root Test therefore gives us no information. However. one can use the Cauchy Condensation Test to see that the series converges.

6 The series L~I In j /3 j is seen to converge by comparing it with the series L~ll/2J. 1 Cauchy Condensation Test Assume thaI CJ > C2 ~ ••• ~ cj ~ ••• O. The series converges if and only if the series converges. Example3.? We apply the Cauchy condensation test to the harmonic series 00 1 j=1 J z:>· It leads us to examine the series Since the latter series diverges. the harmonic series diverges as wen. 8 The series 001 L-:r j=1 J converges if r is a real number that exceeds 1 and diverges if r < 1.