Show that ln ( k + 1 ) − ln k Prove that for k ≥ 1, k ≥ 1, 0 < T k − γ ≤ 1 / k 0 < T k − γ ≤ 1 / k by using the following steps. Now estimate how far T k T k is from γ γ for a given integer k.Since 1 + j ( 1 / 2 ) → ∞, 1 + j ( 1 / 2 ) → ∞, we conclude that the sequence converges. More generally, it can be shown that S 2 j > 1 + j ( 1 / 2 ) S 2 j > 1 + j ( 1 / 2 ) for all j > 1.
Xsection 5.11 series#
Sums and SeriesĪn infinite series is a sum of infinitely many terms and is written in the form We also discuss the harmonic series, arguably the most interesting divergent series because it just fails to converge. This process is important because it allows us to evaluate, differentiate, and integrate complicated functions by using polynomials that are easier to handle. We will use geometric series in the next chapter to write certain functions as polynomials with an infinite number of terms. We introduce one of the most important types of series: the geometric series. We also define what it means for a series to converge or diverge. In this section we define an infinite series and show how series are related to sequences. If you add these terms together, you get a series. We have seen that a sequence is an ordered set of terms.
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Xsection 5.11 pro#
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