(1+3)loge 3+1+322!(log3 3)2 +⋯⋯∞

loge3+loge322+loge333+....∞+3loge3+3loge322+....∞ =eloge3-1+e3loge3-1 =(3-1)+(33-1)=2+26 =28

(1+3)loge 3+1+322!(log3 3)2 +⋯⋯∞

loge3+loge322+loge333+....∞+3loge3+3loge322+....∞ =eloge3-1+e3loge3-1 =(3-1)+(33-1)=2+26 =28

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$(1 + 3) \log_{e} 3 + \frac{1 + 3^{2}}{2!} {(\log_{3} 3)}^{2} + \dots \dots \infty$

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Detailed Solution

$\begin{array}{rcl} [\log_{e}^{3} + \frac{{(\log_{e}^{3})}^{2}}{2} + \frac{{(\log_{e}^{3})}^{3}}{3} + . . . . \infty] + [3 \log_{e}^{3} + \frac{{(3 \log_{e}^{3})}^{2}}{2} + . . . . \infty] \\ = & (e^{\log_{e}^{3}} - 1) + (e^{3 \log_{e}^{3}} - 1) \\ = & (3 - 1) + (3^{3} - 1) = 2 + 26 \\ = & 28 \end{array}$