Final answer:
The student's question concerns the convergence of two series involving logarithmic and arctangent functions. Without more context or specific instructions, one can only suggest general methods like comparison or integral tests to determine the convergence of the series.
Step-by-step explanation:
The student has asked about the convergence or divergence of two series:
- ∑[∞]ₙ₍₃ 1/n.√ℓn n
- ∑[∞]ₙ₈₀₁ τan⁻¹n/n²+1
To assess the convergence of these series, one would typically use comparison tests, integral tests, or other methods of determining series convergence. However, the problem did not specify the methods to use, nor did it provide enough context to solve or simplify the series provided.
For the first series, a comparison to a known convergent or divergent series can be made. Considering the presence of the natural logarithm ℓ(n), which grows slower than n, this is a hint that the series might be compared to something like the harmonic series, though modified by the presence of the square root of ℓ(n).
The second series suggests the use of the alternating series test or a comparison test, noting that the arctangent function is bounded and that 1/(n²+1) will tend to zero as n tends towards infinity.