... x =2andwhenx=−2, the second converges when x = −2, and diverges when x = 2, while the third converges both when x =2andwhenx=−2.These results are all easy to check by direct substitution, and ... harder than it looks; we need to be able to re-arrange power series, and then use the Mean Value Theorem to estimate differences, and show that even when weadd an infinite number of errors, they don’t ... section of the right hand half of the solid outside a cylinder of radiusa and inside the sphere of radius 2a 9972 CHAPTER 7. POWER SERIESProof. We can differentiate n times by 7.9 and we still get...