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.
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The idea of Inequalities begun its improvement from the time whilst C. F. GACSS, A. L. CATCHY and P. L. CEBYSEY, to say simply crucial, laid the theoretical beginning for approximative meth ods. round the finish of the nineteenth and the start of the 20 th century, various inequalities have been proyed, a few of which turned vintage, whereas such a lot remained as remoted and unconnected effects.
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Additional info for A Handbook of Real Variables: With Applications to Differential Equations and Fourier Analysis
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.