10.2 Summing an Infinite Series Feb Do Now Write the following in summation notation 1, ¼, 1/9, 1/16

HW Review

Infinite Series A sum of an infinite number of terms is called an infinite series While it is impossible to add up infinitely many numbers, we can compute the partial sums

Ex Find the 4 th partial sum for the series 1 -1/2 + ¼ + …

Convergence of an Infinite Series An infinite seriesconverges to the sum S if its partial sums converge to S: If the limit does not exist, the series diverges If the limit is infinite, the series diverges to infinity

Note You cannot take the n’th term and determine if that converges – That would tell you if the sequence converges You need to find an expression for the nth partial sum to determine what the series converges to

Ex Investigate the sum numerically then compute the sum S

Linearity of Infinite Series Ifconverge, then: 1) and converge 2) 3)

Geometric Series A geometric series with common ratio is a series defined by a geometric sequence with c and n nonzero Note that |r|<1 is the only instance where the series will converge

Geometric Sums The partial sum: The infinite sums:

Ex Evaluate

Ex Evaluate

Ex Evaluate S =

Divergence Test If the n’th term does not converge to zero, then the seriesdiverges This only tells us if it diverges; if the nth term does converge to 0, we cannot conclude the series will diverge or converge

Ex Show thatdiverges

Ex Determine the convergence or divergence of

Closure What role do partial sums play in defining the sum of an infinite series? HW: p.556 #