Sum of a complex series

Find the sum of $2 n + 1$ terms of the series $3 - 1 + 13 - 9 + 41 - 53 + \dots$

~15 minAnswer checked instantlyrecurring seriessummationgenerating function

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XXIV. Recurring Series, Examples XXIV, No. 13

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Checked against the answer in H. S. Hall & S. R. Knight, Higher Algebra (1891).