Page 47 - Schaum's Outline of Theory and Problems of Advanced Calculus

38 SEQUENCES [CHAP. 2

n
2.70. Prove that ½a n ; b n Š, where a n ¼ð1 þ 1=nÞ and b n ¼ð1 þ 1=nÞ nþ1 ,isa set of nested intervals defining the
number e.
2.71. Prove that every bounded monotonic (increasing or decreasing) sequence has a limit.

2.72. Let fu n g be a sequence such that u nþ2 ¼ au nþ1 þ bu n where a and b are constants. This is called a second
n
order difference equation for u n .(a)Assuming a solution of the form u n ¼ r where r is a constant, prove
2
that r must satisfy the equation r ar b ¼ 0. (b)Use (a)toshow that a solution of the difference
n
n
equation (called a general solution) is u n ¼ Ar 1 þ Br 2 , where A and B are arbitrary constants and r 1 and
2
r 2 are the two solutions of r ar b ¼ 0assumed different. (c)In case r 1 ¼ r 2 in (b), show that a (general)
n
solution is u n ¼ðA þ BnÞr 1 .
2.73. Solve the following difference equations subject to the given conditions: (a) u nþ2 ¼ u nþ1 þ u n , u 1 ¼ 1,
u 2 ¼ 1 (compare Prob. 34); (b) u nþ2 ¼ 2u nþ1 þ 3u n , u 1 ¼ 3, u 2 ¼ 5; (c) u nþ2 ¼ 4u nþ1 4u n , u 1 ¼ 2, u 2 ¼ 8.
Ans.(a) Same as in Prob. 34, (b) u n ¼ 2ð3Þ n 1 þð 1Þ n 1 ðcÞ u n ¼ n 2 n