MAT 21C Lecture Notes - Lecture 4: Telescoping Series, Asteroid Family, Rational Number
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MAT 21C Full Course Notes
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Recap: thm 4: (for nice f ) f (x) l an l. Given a sequence of numbers {an}, an expression of the form an = a1 + a2 + a3 + + an + . The sequence {sn} is the sequence of partial sums of the series. The number sn = k=1 ak is the nth partial sum. Xn=1 (an an+1) = (a1 a2) + (a2 a3) + (a3 a4) + . A geometric series is of the form a + a r + a r2 + + a rn + = a rn 1 = Xm=0 where a 6= 0 is a xed real number and r is a xed ratio (rational number). If |r| < 1 the geometric series converges. If |r| 1 the geometric series diverges. As long as the order of terms is preserved, reindexing a series will not alter its convergence.
