سری زیر را بررسی کنید و مشخص کنید که آیا همگراست: $$ \frac{(-1)^n}{n} $$

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Determine the convergence or divergence by following steps: - step0: Determine the convergence or divergence using the Alternating Series Test: $$\sum _{n=1}^{+\infty}\frac{\left(-1 ight)^{n}}{n}$$ - step1: Find the limit: $$\lim _{n ightarrow +\infty}\left(\left|\frac{\left(-1 ight)^{n}}{n} ight| ight)$$ - step2: Remove the absolute value bars: $$\lim _{n ightarrow +\infty}\left(\frac{1}{n} ight)$$ - step3: Rewrite the expression: $$\frac{1}{\lim _{n ightarrow +\infty}\left(n ight)}$$ - step4: Calculate: $$\frac{1}{+\infty}$$ - step5: Calculate: $$0$$ - step6: Rewrite the expression: $$-\frac{n}{n+1}>-1$$ - step7: Change the sign: $$\frac{n}{n+1}<1$$ - step8: Calculate: $$\frac{n}{n+1}-1<0$$ - step9: Calculate: $$-\frac{1}{n+1}<0$$ - step10: Change the sign: $$\frac{1}{n+1}>0$$ - step11: Rewrite the expression: $$n+1>0$$ - step12: Move the constant to the right side: $$n>0-1$$ - step13: Remove 0: $$n>-1$$ - step14: Calculate: $$\frac{1}{n}$$ - step15: Calculate: $$\frac{1}{n+1}$$ - step16: The inequality is true: $$\frac{1}{n}>\frac{1}{n+1}$$ - step17: Determine the convergence or divergence: $$ ext{Converges}$$ سری $$ \sum_{n=1}^{\infty} \frac{(-1)^n}{n} $$ همگراست. سری $$ \frac{(-1)^n}{n} $$ همگراست.

سری زیر را بررسی کنید و مشخص کنید که آیا همگراست: \( \frac{(-1)^n}{n} \)

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