Determine if the series ∑n=0∞an is convergent or divergent if the partial sum of the n terms of the series is given below. If the series is convergent, determine the value of the series. Sn=5+8n22−7n2

Question

Determine if the series ∑n=0∞an is convergent or divergent if the partial sum of the n terms of the series is given below. If the series is convergent, determine the value of the series.  Sn=5+8n22−7n2

krolaniaN · Accepted Answer

Result on convergence of series: If the sequence of partial sum Sn=a1+a2+…+an of given series ∑n=0∞an is convergent sequence them the series is also convergent . Moreover, the sum of the series is given by ∑n=0∞an=limn→∞Sn Now, the given sequence of partial sum is Sn=5+8n22−7n2 Writing the first few terms of the sequence, &amp;lt;5+82−7,5+8×222−7×22,5+8×232−7×23,…&amp;gt; &amp;lt;−2.6,−1.42,−1.26,…&amp;gt; Clearly, the sequence is monotonically increasing. Also it is bounded above by&#039;0&#039; that is Sn&amp;lt;0,∀n, Hence, the given sequence of partial sum is convergent. In order to get the sum of the series, we will find the limit of the given sequence of partial sum. limn→∞Sn=limn→∞5+8n22−7n2 limn→∞Sn=limn→∞16n−14n =−87 Hence, ∑n=0∞an=limn→∞Sn=limn→∞5+8n22−7n2=−87

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Determine if the series sum_{n=0}^infty a_n is convergent or divergent if the partial sum of the n terms of the series is given below. If the series is convergent, determine the value of the series. S_n=frac{5+8n^2}{2-7n^2}

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