{"id":4308,"date":"2013-12-14T06:05:53","date_gmt":"2013-12-13T21:05:53","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=4308"},"modified":"2014-08-01T19:20:56","modified_gmt":"2014-08-01T10:20:56","slug":"sequences-and-series","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4308","title":{"rendered":"Sequences and series"},"content":{"rendered":"
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A sequence, indicated by u1<\/sub>, u2<\/sub>, …or brief by \\langle u_n \\rangle, is a function defined on the set of natural numbers. The sequence is said to have the limit l or to converge to l, if given any ε > 0 there exists a number N > 0 such that |un<\/sub> – l| < ε for all n > N, and in such case it is described \\lim\\limits_{n \\rightarrow \\infty} u_n =l. If the sequence does not converge, it’s called that it diverges. <\/p>\n

Consider the sequence u1<\/sub>, u1<\/sub> + u2<\/sub>, u1<\/sub> + u2<\/sub> + u3<\/sub>, … or S1<\/sub>, S2<\/sub>, S3<\/sub>, … where Sn<\/sub> = u1<\/sub> + u2<\/sub> + … + un<\/sub>. It’s called \\langle S_n \\rangle the sequence of partial sums of the sequence \\langle u_n \\rangle. The symbol <\/p>\n

\\displaystyle u_1 + u_2 + u_3 + \\cdots or \\displaystyle \\sum_{n=1}^{\\infty}u_n or briefly \\displaystyle \\sum u_n<\/p>\n

is defined as synonymous with \\langle S_n \\rangle and is called an infinite series. This series will converge or diverge according as \\langle S_n \\rangle converges or diverges. If it converges to S it’s called S as the sum of the series. <\/p>\n

The following are some important theorems concerning infinite series. <\/p>\n

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  1. The series \\displaystyle \\sum_{n=1}^{\\infty}\\frac{1}{n^p} converges if p > 1 and diverges if p ≤ 1.<\/li>\n
  2. If ∑|un<\/sub>| converges and |vn<\/sub>| ≤ |un<\/sub>|, then ∑|vn<\/sub>| converges.<\/li>\n
  3. If ∑|un<\/sub>| converges, then ∑un<\/sub> converges. <\/li>\n
  4. If ∑|un<\/sub>| diverges and vn<\/sub> ≥ |un<\/sub>|, then ∑vn<\/sub> diverges. <\/li>\n
  5. The series ∑|un<\/sub>|, where |un<\/sub>| = f(n) ≥ 0, converges or diverges according as \\displaystyle \\int_{1}^{\\infty}f(x)dx = \\lim\\limits_{M \\rightarrow \\infty}\\int_{1}^{M}f(x)dx exists or does not exist. This theorem is often called the integral test. <\/li>\n
  6. The series ∑|un<\/sub>| diverges if \\displaystyle \\lim\\limits_{n \\rightarrow \\infty}|u_n| \\neq 0. However, if \\displaystyle \\lim\\limits_{n \\rightarrow \\infty}|u_n| = 0 the series may or may not converge.<\/li>\n
  7. Suppose that \\displaystyle \\lim\\limits_{n \\rightarrow \\infty}\\left|\\frac{u_{n+1}}{n_n}\\right| = r. Then the series ∑un<\/sub> converges (absolutely) if r < 1 and diverges if r > 1. If r = 1, no conclusion can be drawn. This theorem is often referred to as the ratio test.<\/li>\n<\/ol>\n<\/blockquote>\n