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I'm sure the list you seek would be almost endless. One may suggest that you browse the books by Bruce C. Berndt, Ramanujan's Notebooks, Part I, II, etc, Springer.

Euler's formula $e^{\pi i}+1=0$ is everyone's favorite. In the same spirit, but to show the massive computational power of Ramanujan, here is special case from Entry 17, page 435, Part V, of the above-mentioned series. $$\sum_{k=1}^{\infty}\frac{(-1)^{k-1}k}{e^{k\pi\sqrt{3}}-(-1)^k}=\frac1{4\pi\sqrt{3}}-\frac1{24}.$$ Notice the interplay of the two famous constants $\pi$ and $e$.

In view of Robert Israel's reasonable comment, perhaps we could go for the modest expressions: $$\sqrt{2\left(1-\frac1{3^2}\right)\left(1-\frac1{7^2}\right)\left(1-\frac1{11^2}\right)\left(1-\frac1{19^2}\right)} =\left(1+\frac17\right)\left(1+\frac1{11}\right)\left(1-\frac1{19}\right)$$ found in S. Ramanujan, Notebooks of Srinivasa Ramanujan, Volume II, Tata Institute of Fundamental Research, Bombay, 1957. See pp. 309 and 363.

Regarding the second example, it is feasible to encourage students to find similar results of the same kind because there are many. They would be able experiment.

I'm sure the list you seek would be almost endless. One may suggest that you browse the books by Bruce C. Berndt, Ramanujan's Notebooks, Part I, II, etc, Springer.

Euler's formula $e^{\pi i}+1=0$ is everyone's favorite. In the same spirit, but to show the massive computational power of Ramanujan, here is special case from Entry 17, page 435, Part V, of the above-mentioned series. $$\sum_{k=1}^{\infty}\frac{(-1)^{k-1}k}{e^{k\pi\sqrt{3}}-(-1)^k}=\frac1{4\pi\sqrt{3}}-\frac1{24}.$$ Notice the interplay of the two famous constants $\pi$ and $e$.

In view of Robert Israel's reasonable comment, perhaps we could go for the modest expressions: $$\sqrt{2\left(1-\frac1{3^2}\right)\left(1-\frac1{7^2}\right)\left(1-\frac1{11^2}\right)\left(1-\frac1{19^2}\right)} =\left(1+\frac17\right)\left(1+\frac1{11}\right)\left(1+\frac1{19}\right)$$ found in S. Ramanujan, Notebooks of Srinivasa Ramanujan, Volume II, Tata Institute of Fundamental Research, Bombay, 1957. See pp. 309 and 363.

Regarding the second example, it is feasible to encourage students to find similar results of the same kind because there are many. They would be able experiment.

I'm sure the list you seek would be almost endless. One may suggest that you browse the books by Bruce C. Berndt, Ramanujan's Notebooks, Part I, II, etc, Springer.

Euler's formula $e^{\pi i}+1=0$ is everyone's favorite. In the same spirit, but to show the massive computational power of Ramanujan, here is special case from Entry 17, page 435, Part V, of the above-mentioned series. $$\sum_{k=1}^{\infty}\frac{(-1)^{k-1}k}{e^{k\pi\sqrt{3}}-(-1)^k}=\frac1{4\pi\sqrt{3}}-\frac1{24}.$$ Notice the interplay of the two famous constants $\pi$ and $e$.

In view of Robert Israel's reasonable comment, perhaps we could go for the modest expressions: $$\sqrt{2\left(1-\frac1{3^2}\right)\left(1-\frac1{7^2}\right)\left(1-\frac1{11^2}\right)\left(1-\frac1{19^2}\right)} =\left(1+\frac17\right)\left(1+\frac1{11}\right)\left(1-\frac1{19}\right)$$ found in S. Ramanujan, Notebooks of Srinivasa Ramanujan, Volume II, Tata Institute of Fundamental Research, Bombay, 1957. See pp. 309 and 363.

I'm sure the list you seek would be almost endless. One may suggest that you browse the books by Bruce C. Berndt, Ramanujan's Notebooks, Part I, II, etc, Springer.

Euler's formula $e^{\pi i}+1=0$ is everyone's favorite. In the same spirit, but to show the massive computational power of Ramanujan, here is special case from Entry 17, page 435, Part V, of the above-mentioned series. $$\sum_{k=1}^{\infty}\frac{(-1)^{k-1}k}{e^{k\pi\sqrt{3}}-(-1)^k}=\frac1{4\pi\sqrt{3}}-\frac1{24}.$$ Notice the interplay of the two famous constants $\pi$ and $e$.

In view of Robert Israel's reasonable comment, perhaps we could go for the modest expressions: $$\sqrt{2\left(1-\frac1{3^2}\right)\left(1-\frac1{7^2}\right)\left(1-\frac1{11^2}\right)\left(1-\frac1{19^2}\right)} =\left(1+\frac17\right)\left(1+\frac1{11}\right)\left(1-\frac1{19}\right)$$ found in S. Ramanujan, Notebooks of Srinivasa Ramanujan, Volume II, Tata Institute of Fundamental Research, Bombay, 1957. See pp. 309 and 363.

Regarding the second example, it is feasible to encourage students to find similar results of the same kind because there are many. They would be able experiment.

I'm sure the list you seek would be almost endless. One may suggest that you browse the books by Bruce C. Berndt, Ramanujan's Notebooks, Part I, II, etc, Springer.

Euler's formula $e^{\pi i}+1=0$ is everyone's favorite. In the same spirit, but to show the massive computational power of Ramanujan, here is special case from Entry 17, page 435, Part V, of the above-mentioned series. $$\sum_{k=1}^{\infty}\frac{(-1)^{k-1}k}{e^{k\pi\sqrt{3}}-(-1)^k}=\frac1{4\pi\sqrt{3}}-\frac1{24}.$$ Notice the interplay of the two famous constants $\pi$ and $e$.

I'm sure the list you seek would be almost endless. One may suggest that you browse the books by Bruce C. Berndt, Ramanujan's Notebooks, Part I, II, etc, Springer.

Euler's formula $e^{\pi i}+1=0$ is everyone's favorite. In the same spirit, but to show the massive computational power of Ramanujan, here is special case from Entry 17, page 435, Part V, of the above-mentioned series. $$\sum_{k=1}^{\infty}\frac{(-1)^{k-1}k}{e^{k\pi\sqrt{3}}-(-1)^k}=\frac1{4\pi\sqrt{3}}-\frac1{24}.$$ Notice the interplay of the two famous constants $\pi$ and $e$.

In view of Robert Israel's reasonable comment, perhaps we could go for the modest expressions: $$\sqrt{2\left(1-\frac1{3^2}\right)\left(1-\frac1{7^2}\right)\left(1-\frac1{11^2}\right)\left(1-\frac1{19^2}\right)} =\left(1+\frac17\right)\left(1+\frac1{11}\right)\left(1-\frac1{19}\right)$$ found in S. Ramanujan, Notebooks of Srinivasa Ramanujan, Volume II, Tata Institute of Fundamental Research, Bombay, 1957. See pp. 309 and 363.