10_Infinite_Series(1)(1).pdf - k n=1 n=1 a Infinite Series an = klim \u2192 n Does it make sense to even think about the sum of an infinite number of terms

10_Infinite_Series(1)(1).pdf - k n=1 n=1 a Infinite Series...

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Infinite Series: 1 1 lim k k n n n n a a → = = = Does it make sense to even think about the sum of an infinite number of terms? 1 ) An infinite series is a sum of the terms of an infinite sequence. A series is denoted by 1 n n a = but we frequently write n a The notation is an abbreviation because it really means: 1 1 2 3 n n a a a a = = + + + Example 1 ] Write the sum 1 1 2 n n = in explicit form. Solution : 1 2 3 4 1 1 1 1 1 1 2 2 2 2 2 n n = = + + + + Example 2 ] Write the sum 2 1 1 1 n n n = in explicit form. Solution : 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 3 3 4 4 5 n n n = = + + + + Example 3 ] Write the sum 0 1 1 n n = + in explicit form. Solution : 0 ( 1) 1 1 1 1 1 1 2 3 4 n n n = = + + + 2 ) Sequence of Partial Sums. Convergence and Divergence of Infinite Series Given a series 1 1 2 3 n n a a a a = = + + + , its partial sum is denoted by n s and 1 1 2 3 k n n n k a a a a a s = = = + + + + That is, 1 1 1 1 k k a a s = = = ; 1 2 2 1 2 k k a a a s = = = + ; 1 3 3 1 2 3 k k a a a a s = = = + + ; 1 4 4 1 2 3 4 k k a a a a a s = = = + + + ; 1 5 5 5 1 2 3 4 k k a a a a a a s = = = + + + + ; etc… If the sequence n s is convergent and lim n n s s → = exists as a real number , then 1 n n a = is convergent . We write 1 1 2 3 n n a a a a s = = + + + = where s is the sum of the series. If the sequence n s is divergent, then 1 n n a = is divergent.