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### Section 10.3 : Series - Basics

For problems 1 – 3 perform an index shift so that the series starts at $$n = 3$$.

1. $$\displaystyle \sum\limits_{n = 1}^\infty {\left( {n{2^n} - {3^{1 - n}}} \right)}$$ Solution
2. $$\displaystyle \sum\limits_{n = 7}^\infty {\frac{{4 - n}}{{{n^2} + 1}}}$$ Solution
3. $$\displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 3}}\left( {n + 2} \right)}}{{{5^{1 + 2n}}}}}$$ Solution
4. Strip out the first 3 terms from the series $$\displaystyle \sum\limits_{n = 1}^\infty {\frac{{{2^{ - n}}}}{{{n^2} + 1}}}$$. Solution
5. Given that $$\displaystyle \sum\limits_{n = 0}^\infty {\frac{1}{{{n^3} + 1}}} = 1.6865$$ determine the value of $$\displaystyle \sum\limits_{n = 2}^\infty {\frac{1}{{{n^3} + 1}}}$$. Solution