For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges. ∑k=1∞10k

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For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges.  ∑k=1∞10k

Theodore Schwartz · Accepted Answer

Given, the series is ∑k=1∞10k We have to find the first four terms of the sequence of partial sums, make a conjecture and state that the series is divergent. 1. If ∑ak is an infinite series, then Sk=a1+a2+a3+...+ak is called the n partial sum 2.If ∑ak is an series of positive terms s.t. limk→∞akak+1=l, then the series ∑ak is divergent if l&amp;lt;1 Now, to find the first four terms of a sequence of partial sums ak=10k S1=a1=101=10 S2=a1+a2=101+102=10+100=110 S3=a1+a2+a3=101+102+103=10+100+1000=1110 S4=a1+a2+a3+a4=101+102+103+104=10+100+1000+10000=11110 Since S1=10 S2=110 S3=1110 S4=11110 ⋮ Sk=∑k=0∞10k→∞ Now, ak=10k, ak+1=10k+1 limk→∞10k10k+1=lim110=110&amp;lt;1 Hence the series is divergent.

Jeffrey Jordon · Accepted Answer

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For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges. sum_{k=1}^infty10^k

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