“My name is Alice. What’s yours?”
“I’m Mercury; welcome to my wedding.”
“Wedding?! You seem so young to be getting married!”
“I know, I’m supposed to look young, but actually I’m very old.” Mercury looked around as if he was about to slip out a secret. “All my family have been telling me for centuries to get married, settle down, raise some kids, but I just want to fly away.”
“Me too! I love to fly!!” Alice felt as if her words had just echoed around the whole universe and that everybody had heard them. She blushed a little.
At that moment, a thunderous clap jolted both Alice and Mercury from their reverie and they both turned their head to stare at what appeared to be the Sun slowly coming down to land in the middle of the... party! Alice had been so charmed by her new friend that she hadn’t noticed that she was in the middle of some kind of carnival.
As the figure descended from the heavens it became clear that this was no ordinary Sun, but a woman who shone like the brightest thing in the world. She was cloaked in a long gown the colour of the evening sky and embroidered with symbols and numbers.
“Who... is... that?” asked Alice in complete awe of the vision before her. Everyone bowed towards this special lady and Alice copied them.
“That is Arithmetica.” Explained Mercury; “She is the one, the two, the three and more. She is the infinite and the nothing. Even the gods bow down before her because without her they too would be nothing.” Mercury looked around, as if he was about to leave. “I think she has also brought her sisters along to remind us all about how important they are.”
As if in response to this whispered insult, Arithmetica was about to put on a show of power. From her forehead she emanated a dense beam of bright light. In mid-air the beam split into two as if forming two perpendicular axes. The light quickly split further in all directions until a web of light filled the entire wedding party. Then in the blink of an eye the whole lacework of laser beams contracted back into her forehead.
Alice thought her jaw was resting on the ground and tried to compose herself. For once in her life, she was quite literally speechless.
“Look, Alice, I hope you can stay for the wedding, but I really must go and take my official place. Let me guide you to a front seat.” Mercury took Alice’s hand and they glided towards the very bizarre-looking guests.
“Here, you can sit next to Pythagoras. You can talk to him about music.” And with that, Mercury flew away to take his place next to his wife-to-be, who immediately started nagging him. Poor Mercury! thought Alice.
Find the sum of the following infinite series:
You must include a valid method as well as the final sum.
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