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∑n=3∞4n+2ln⁡nnln⁡n\sum_{n=3}^{\infty}\frac{4n+2\ln n}{n \ln n} Determine if This series is Convegent or Divergent by using Direct comparison test.

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Q. ∑n=3∞4n+2ln⁡nnln⁡n\sum_{n=3}^{\infty}\frac{4n+2\ln n}{n \ln n} Determine if This series is Convegent or Divergent by using Direct comparison test.
  1. Identify Comparison Series: Identify a simpler comparison series: We compare (4n+2ln⁡n)/(nln⁡n)(4n + 2\ln n)/(n \ln n) to 4/n4/n, since as n→∞n \rightarrow \infty, the term 2ln⁡n/(nln⁡n)2\ln n/(n \ln n) becomes negligible.
  2. Apply Direct Comparison Test: Apply the Direct Comparison Test:\newlineWe know that 4n\frac{4}{n} is a p-series with p=1p = 1, which is known to diverge.
  3. Compare to 4n\frac{4}{n}: Compare the original series to 4n\frac{4}{n}:\newlineSince 4n+2ln⁡nnln⁡n≥4n\frac{4n + 2\ln n}{n \ln n} \geq \frac{4}{n} for all n≥3n \geq 3, and since 4n\frac{4}{n} diverges, by the Direct Comparison Test, the original series also diverges.

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