Show that the series is convergent. how many terms of the series do we need to add in order to find the sum to the indicated accuracy? sum_(n=1)^(infinity) (-1)^(n+1)/( n^7)text( ) \(|text(error)| < 0.5 \)

For a convergent alternating series with value and th partial sums denoted by , the th error is bounded by the absolute value of the th term's absolute value:



We have



so in order to have an error within 0.00005 of the sum's actual value, we need terms such that






which suggests that we require terms at the least to approximate the series within the given accuracy.