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# Iv 78 positive integral variable 175 examples xxx 1

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Appendix I.

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[IV : 78] POSITIVE INTEGRAL VARIABLE 175 Examples XXX. 1. The series r m + r m +1 + . . . is convergent if - 1 < r < 1, and its sum is 1 / (1 - r ) - 1 - r - · · · - r m - 1 ( § 77 , (2)). 2. The series r m + r m +1 + . . . is convergent if - 1 < r < 1, and its sum is r m / (1 - r ) ( § 77 , (4)). Verify that the results of Exs. 1 and 2 are in agreement. 3. Prove that the series 1 + 2 r + 2 r 2 + . . . is convergent, and that its sum is (1 + r ) / (1 - r ), ( α ) by writing it in the form - 1 + 2(1 + r + r 2 + . . . ), ( β ) by writing it in the form 1 + 2( r + r 2 + . . . ), ( γ ) by adding the two series 1 + r + r 2 + . . . , r + r 2 + . . . . In each case mention which of the theorems of § 77 are used in your proof. 4. Prove that the ‘arithmetic’ series a + ( a + b ) + ( a + 2 b ) + . . . is always divergent, unless both a and b are zero. Show that, if b is not zero, the series diverges to + or to -∞ according to the sign of b , while if b = 0 it diverges to + or -∞ according to the sign of a . 5. What is the sum of the series (1 - r ) + ( r - r 2 ) + ( r 2 - r 3 ) + . . . when the series is convergent? [The series converges only if - 1 < r 5 1. Its sum is 1, except when r = 1, when its sum is 0.] 6. Sum the series r 2 + r 2 1 + r 2 + r 2 (1 + r 2 ) 2 + . . . . [The series is always convergent. Its sum is 1 + r 2 , except when r = 0, when its sum is 0.] 7. If we assume that 1 + r + r 2 + . . . is convergent then we can prove that its sum is 1 / (1 - r ) by means of § 77 , (1) and (4). For if 1+ r + r 2 + · · · = s then s = 1 + r (1 + r 2 + . . . ) = 1 + rs. 8. Sum the series r + r 1 + r + r (1 + r ) 2 + . . .
[IV : 78] LIMITS OF FUNCTIONS OF A 176 when it is convergent. [The series is convergent if - 1 < 1 / (1 + r ) < 1, i.e. if r < - 2 or if r > 0, and its sum is 1 + r . It is also convergent when r = 0, when its sum is 0.] 9. Answer the same question for the series r - r 1 + r + r (1 + r ) 2 - . . . , r + r 1 - r + r (1 - r ) 2 + . . . , 1 - r 1 + r + r 1 + r 2 - . . . , 1 + r 1 - r + r 1 - r 2 + . . . . 10. Consider the convergence of the series (1 + r ) + ( r 2 + r 3 ) + . . . , (1 + r + r 2 ) + ( r 3 + r 4 + r 5 ) + . . . , 1 - 2 r + r 2 + r 3 - 2 r 4 + r 5 + . . . , (1 - 2 r + r 2 ) + ( r 3 - 2 r 4 + r 5 ) + . . . , and find their sums when they are convergent. 11. If 0 5 a n 5 1 then the series a 0 + a 1 r + a 2 r 2 + . . . is convergent for 0 5 r < 1, and its sum is not greater than 1 / (1 - r ). 12. If in addition the series a 0 + a 1 + a 2 + . . . is convergent, then the series a 0 + a 1 r + a 2 r 2 + . . . is convergent for 0 5 r 5 1, and its sum is not greater than the lesser of a 0 + a 1 + a 2 + . . . and 1 / (1 - r ). 13. The series 1 + 1 1 + 1 1 · 2 + 1 1 · 2 · 3 + . . . is convergent. [For 1 / (1 · 2 . . . n ) 5 1 / 2 n - 1 .] 14. The series 1 + 1 1 · 2 + 1 1 · 2 · 3 · 4 + . . . , 1 1 + 1 1 · 2 · 3 + 1 1 · 2 · 3 · 4 · 5 + . . . are convergent. 15. The general harmonic series 1 a + 1 a + b + 1 a + 2 b + . . . , where a and b are positive, diverges to + . [For u n = 1 / ( a + nb ) > 1 / { n ( a + b ) } . Now compare with 1 + 1 2 + 1 3 + . . . .]

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[IV : 79] POSITIVE INTEGRAL VARIABLE 177 16. Show that the series ( u 0 - u 1 ) + ( u 1 - u 2 ) + ( u 2 - u 3 ) + . . . is convergent if and only if u n tends to a limit as n → ∞ . 17. If u 1 + u 2 + u 3 + . . . is divergent then so is any series formed by grouping the terms in brackets in any way to form new single terms. 18. Any series, formed by taking a selection of the terms of a convergent series of positive terms, is itself convergent.
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